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THICK-LENS  OPTICS 

AN  ELEMENTARY  TREATISE 

FOR  THE   STUDENT  AND 

THE  AMATEUR 


BY 

ARTHUR   LATHAM   BAKER,    PH.D. 

MANUAL  TRAINING   HIGH   SCHOOL,    BROOKLYN,    N.Y. 

Author  of 
"QUATERNIONS  AS  THE  RESULT  OF  ALGEBRAIC  OPERATIONS' 


ILLUSTRATED 


NEW  YORK 

D.  VAN  NOSTRAND   COMPANY 

25   PARK   PLACE 
1912 


Copyright,  1912 

BY 

D.  VAN  NOSTRAND  COMPANY 


THE-PLIMPTON-PRESS-NORWOOD-MASS-U-S-A 


PREFACE 

THIS  volume  is  the  outcome  of  an  attempt  to  answer 
certain  questions  regarding  the  optics  of  the  microscope 
and  telescope;  questions  to  which  no  thoroughly  satisfac- 
tory answers  could  be  found  in  any  literature  accessible 
to  the  author. 

Many  answers  were  found,  but  they  were  discordant 
and  unusable  for  practical  work,  mainly  by  reason  of  their 
complexity  and  seeming  contradictoriness  and  lack  of 
co-ordination. 

The  following  pages  seek  to  answer  these  questions  in  a 
manner  so  plain  and  simple  that  the  average  amateur  can 
find  out  for  himself  what  is  going  on  optically  in  his  camera, 
microscope,  or  telescope. 

To  this  end  the  mathematics  is  of  the  simplest  kind,  so 
that  the  busy  man  who  has  forgotten  all  or  most  of  his 
mathematics  can  nevertheless  work  his  way  through,  pro- 
vided he  can  use  the  simplest  kind  of  algebra,  two  theorems 
in  elementary  geometry  and  one  in  trigonometry.  For 
the  reader  who  has  not  had  trigonometry,  the  few  simple 
principles  required  are  given  in  the  text.  So  far  as  mathe- 
matical difficulties  go,  any  high-school  student  is  sufficiently 
equipped. 

As  an  aid  to  concreteness  and  clearness  the  investigations 
are  based  upon  graphic  principles  as  much  as  possible  and 
along  intuitive  lines. 

For  the  more  inquisitive  reader  who  desires  a  more 

iii 


271628 


iv  PREFACE 

rigidly  logical  basis  one  investigation  is  given  in  analytical 
form,  as  a  supplement  to  the  preceding  intuitive  ones. 
This  can  safely  be  omitted  by  those  not  interested,  without 
destroying  the  continuity  of  the  text. 

The  text  is  a  working  one,  intended  to  give  the  reader 
practical  and  intelligible  rules  of  procedure,  with  full  and 
thorough  explanations,  so  that  the  most  cursory  reader  can 
utilize  them.  Many  practical  examples  are  fully  worked 
out  and  many  more  given  for  practice. 

Particular  pains  has  been  taken  to  reconcile  seemingly 
contradictory  formulae  for  the  same  result,  which,  unrecon- 
ciled, leave  the  reader  in  the  deepest  uncertainty,  the  fault 
of  most  of  the  literature  on  the  subject. 

This  volume,  for  the  first  time  apparently,  assembles 
these  rules,  answers,  and  formulae  in  one  consistent  whole, 
in  a  practical  form  intelligible  to  the  non-technical  reader. 

The  formulation  of  the  methods  of  procedure  is  so  stand- 
ardized and  simplified  (§  106)  that  it  is  expected  that  the 
reader  can  readily  utilize  the  necessary  calculations,  con- 
cretely visualized  and  checked  by  the  graphic  constructions 
(§  107). 

His  ability  to  do  so  ought  to  render  his  use  of  optical 
instruments  that  much  more  intelligent  and  interesting, 
and  enable  him  to  know  roughly  how  his  instrument  is 
doing  its  work,  what  effect  a  change  of  lens  or  of  its  position 
would  have,  how  to  decide  in  a  rough  way  what  form  of 
lens  he  wants  for  certain  effects,  and  how  he  could  modify 
those  effects. 

The  investigation  is  for  a  single  monochromatic  ray,  and 
therefore  the  questions  of  achromatic  and  spherical  aber- 
ration are  not  touched  upon,  as  not  being  within  the  scope 
of  the  simple  treatment  used. 

The  goal  of  the  work  is,  of  course,  the  practical  calcula- 


PREFACE  V 

tions  of  §§  106,  107  and  Chapter  V,  though  many  other 
important  calculations  are  gathered  on  the  way. 

To  render  the  work  less  an  isolated  monograph  and  make 
it  more  useful  to  the  general  reader,  a  number  of  sections 
have  been  added,  to  round  out  the  subject  somewhat 
toward  the  nature  of  a  handbook  and  to  increase  its 
practical  value  to  the  owner  of  an  optical  instrument; 
not  the  least  valuable  of  which  will  be  the  chapter  on 
Experimental  Observations.  This  chapter  will  enable  the 
reader  to  get  an  experimental  acquaintance  with  the  optical 
constants  of  his  lenses. 

The  author  makes  little  claim  to  novelty,  except  in  the 
simplification  and  workability  of  the  rules  of  procedure. 

A.  L.  B. 

BROOKLYN,  N.  Y. 
October  1,  1912. 


CONTENTS 


PAGE 

CHAPTER  I 

Surface  Refraction 1 

Construction  for 3 

Equation  for 4 

CHAPTER  II 

Thin  Lenses,  Equation 6 

Direction  of  Light 8 

Optical  Center 9 

Diagrammatic  Investigations 

Positive  Lens 10 

Negative  Lens 12 

Oblique  Rays 15 

Parallel  Rays 17 

Use  of  Formulae 18 

Graphic  Check 22 

Diopters 25 

Spectacles 26 

Magnification 28 

Copying 31 

Exposure 33 

Hyperfocal  Distance 34 

Magnifying  Power 36 

CHAPTER  III 

Thick  Lenses 39 

Principal  Points 40 

Nodal  Points 42 

Optical  Center 43 

Construction  for  Nodal  Points 44 

Image  in  Nodal  Plane 47 

vii 


viii  CONTENTS 

PAGE 

Focal  Length 49 

Construction  for  Image 53 

Use  of  Formulae 54 

Graphic  Tracing  of  Ray 54 

Analytical  Investigation 55 

General  Equation 57 

CHAPTER  IV 

Combinations  of  Lenses 64 

Thin  Lenses  in  Contact 64 

Thin  Lenses  not  in  Contact 65 

Back  Focal  Distance:  Thin  Lenses 67 

Equivalent  Focus:  Thin  Lenses 67 

Back  Focal  Distance:  Light  from  Right.      ......  70 

Back  Focal  Distance:  Thick  Lenses .  71 

Equivalent  Focal  Distance:  Thick  Lenses 71 

Nodal  Distance 72 

Resume 73 

Use  of  Formulae,  etc 74 

Graphic  Construction 76 

Magnifying  Power,  Microscope 87 

Telescope 88 

Opera  Glass 89 

CHAPTER  V 

Telephoto  Lens 90 

Focal  Length / 91 

Telephoto  Magnification 91 

Focal  Distances 92 

Image  Distance 94 

Focal  Radius 94 

Object  Distance  for  given  Magnification 96 

Reduction  Factor      .      .      ...     .     . '    •*     .  -  .     .      .      .  '  97 

CHAPTER  VI 

Reflection  at  Surfaces 98 

Graphic  Construction 99 

CHAPTER  VII 

Experimental  Observations .  102 


CONTENTS  ix 

PAGE 

Radius  of  Curvature  of  Surface 102 

Radius  of  Curvature  of  Surface:  Small  Radius       ....  104 

Focal  Length  of  Thin  Positive  Lens 

With  Sun 104 

Lens  Distances 105 

With  a  Telescope 105 

Different  Lens  Positions 105 

Equality  of  Object  and  Image 105 

Comparison  of  Images 105 

Focal  Length  of  Thick  Positive  Lens 

Highly  Magnified  Image 107 

Swing  of  Camera        107 

Movement  of  Screen 108 

Angle  of  Vision 108 

Unit-Screen  Movement 108 

Measurement  of  Image 109 

Comparison  with  Standard  Lens 109 

Double  Focus 110 

Lens  Displacement Ill 

Focal  Radius  of  Negative  Lens 

With  Sun Ill 

With  Stronger  Positive  Lens 113 

With  Positive  Lens  and  Comparison  of  Images        .      .    «.  114 

Location  of  Nodals 114 

Magnifying  Power 

Telescope 115 

Visual  Comparison  of  Images 115 

Microscope 

Visual  Comparison  of  Images 116 

Distribution  between  Objective  and  Ocular     .      .      .      .  116 

Index  of  Refraction 118 

Practical  Suggestions 121 


THICK-LENS  OPTICS 

CHAPTER  I 

SURFACE  REFRACTION 

1.  As  this  is  intended  as  a  working  manual,  and  not  a 
treatise,  it  will  be  assumed  at  the  outset  that  the  reader 
is  acquainted  with  the  fundamental  principles  of  optical 
refraction,  the  commonplaces  of  the  elementary  text-books, 
viz.: 

(a)    Light  rays  are  propagated  in  straight  lines. 

(6)  In  passing  from  one  medium  to  another,  a  ray  of 
light  is  deflected  towards  the  normal  to  the  surface  in 
passing  into  the  denser  medium;  vice  versa  in  passing  out. 

2.  Definition.  —  The  ratio  of  rise  to  slant  of  a  line  is 
called  the  sine  of  the  angle  of  inclination;  of  rise  to  run, 
the  tangent  of  the  angle  of  inclination. 

For  example:  In  a  roof  whose  vertical  height  (rise)  is 
3  feet,  whose  half  width  (run)  is  4,  the  length  of  the  rafters 
(slant)  will  be  5,  and  f  is  the  sine  of  the  angle  of  inclination 
of  the  roof  to  the  horizon;  f  is  the  tangent  (generally  written 
tan)  of  the  angle  of  inclination,  ratio  of  rise  to  run. 

3.  Definition.  —  The  ratio  of  the  sine  of  the  angle  of 
incidence  to  the  sine  of  the  angle  of  refraction  is  called  the 
index  of  refraction  for  the  denser  medium,  the  ray  passing 
into  the  denser  medium  from  air. 

1 


2  THICK-LENS    OPTICS 

4.   Surface  Refraction. 


8  Jingle  of  incidence 
Emergent  Tay 


Diagram  showing  angles  of  incidence,  refraction,  etc. 

The  index  of  refraction  is  generally  indicated  by  the 
Greek  letter  p,  thus: 

0  sin  (3 


A*  = 


sn 


sn  a 


Unless  otherwise  specified,   we  will  consider  only  two 
media:  air  and  glass. 

5.  Law  of  Refraction  of  Light.  —  The  index  of  refraction 
for  the  same  two  media  is  constant,  whatever  the  angle  of 
incidence. 

6.  Trigonometric  Law  of  Sines.  —  In  any  triangle,  the 
sides  are  proportional  to  the  sines  of  the  opposite  angles. 

Proof.     By  definition  of  sine, 


sin  A  =  r 


r»          «* 

sin  B  =  - 
a 


Whence,  by  division, 


sin  A       a 

=  =  r         Q.E.D. 

sin  B       b 


SURFACE  REFRACTION  3 

7.  Note.  —  In  all  construction  diagrams,  the  order  of 
construction  is  indicated  by  the  alphabetical  order  of  the 
letters;  e.g.  in  the  diagram  below,  draw  the  circle  a  with 
the  radius  indicated,  then  draw  the  circle  6,  then  locate 
the  point  C,  then  locate  the  point  Z),  and  then,  since  this 
ends  the  series  of  letters,  draw  the  refracted  ray  as  shown. 

8.  Construction  for  Surface  Refraction. 


'a,with  radius  k 
b,witk  radius  m 


surface  at 
point  of  in- 
cidence 


Diagram  showing  construction  for  surface  refraction.     Given  the 
incident  ray,  to  find  the  refracted  ray. 

For  refraction  out,  interchange  the  letters  C  and  D. 

9.  Definition.  —  Any  arbitrary  line  through  the  center 
of  curvature  is  called  the  axis  of  the  surface.     The  point 
where  it  pierces  the  surface  is  called  the  vertex. 

10.  Convention  as  to  Signs.  —  Distances  measured  to 
the  right  from  the  vertex  are  considered  as  positive;  those 
to  the  left  negative. 

Note.  —  In  the  following  investigations,   the  diagrams 
will  usually  be  so  taken  as  to  make  all  the  elements  con- 


4  THICK-LENS    OPTICS 

sidered  positive.  This  will  give  normal  equations  which 
may  be  considered  typical  for  all  cases. 

11.  Equation  for  Surface  Refraction  (incident  onto 
denser  medium). 

By  the  Law  of  Sines  (§  6), 

sin  QOR  =  sin  <*» 


=  n  sin  <£ 

sin  <f>   _  sine  of  incident  angle 
sin  <£'  ~  sine  of  refracted  angle 


P    F-limit  ofP        Q  Source 
ofroA) 


Jrri 


sin  FOR  =  sin 


.  sin  QOR  =  /*         sin  FOR 


QO-PR  =  n-PO-QR 

[sin  QOR  =  sin  FOR 

As  R  approaches  S,  P  approaches  some  point  F  as  a 
limit. 

.'.     (u  —  r)  w  =  /«.  (w  —  r)  u 

or,  in  the  more  usual  form, 


SURFACE   REFRACTION 


w      u  r 

~~u  =  dist.  from  vertex  to  source,  +  to  right 
w  =  dist.  to  image,  +  to  right,  —  to  left 
r  =  radius  of  curvature,  +  to  right 
fi.  =  index  of  refraction1 

12.   When  the  source  is  very  distant,  i.e.  u  =  oo,  w  takes 
the  special  value  /,  and 

/  =  distance  to  point  through  which  rays  parallel  to  the 
axis  in   the   rarer   medium   meet   the   axis   after 
refraction 
=  focal  radius  of  the  surface  for  parallel  entering  rays 

(r  meas.  from  vertex  to  right  is  pos.  and  vice 


versa). 

N.  B.  —  This  formula  holds  whether  light  comes  from 
right  or  left,  or  surface  convex  or  concave. 

—  T 

13.  For  emergent  rays  /0  =  —  ^r  =  focus  for  rays  par- 

allel in  the  denser  medium.   Note  how  it  differs  from  that 
of  §  12. 

14.  Graphic   Check.  —  Check  the   calculation  of  /  by 
similar  triangles  drawn  to  scale,  „ 

in  which  the  sides  are  repre- 
sented  as  shown.  This  will 
readily  detect  large  errors  of  cal- 
culation in  time  to  prevent  their  vitiating  later  calculations. 

1  The  /*  represents  the  ratio  of  the  sine  of  the  angle  of  incidence  to 
the  sine  of  the  angle  of  refraction.  In  the  case  of  incidence  on  a  denser 
medium  it  will  be  the  index  of  refraction  and  an  improper  fraction, 
but  in  the  case  of  incidence  on  a  rarer  medium  it  will  be  the  reciprocal 
of  the  index  of  refraction  and  a  proper  fraction. 


CHAPTER  II 

THIN  LENSES 

15.  By  thin  lenses  is  meant  lenses  whose  thickness  can 
be  practically  neglected  in  comparison  with  the  other 
elements  under  consideration. 


By  §  11,  for  the  first  surface, 

d  —  -  =  P  ~~  1  =  £ 
w       u  r  f 

Similarly,  for  the  second  surface,  the  image  of  the  first 
surface  being  the  object  of  the  second  surface,  and  the 
index  for  the  second  surface  being  inverted,  because  the 
ray  is  emergent  instead  of  incident  (see  §  11,  footnote), 

1  1 

/*       _1_  _  /M.  FFor  significance  of  the  letters 

v        w'  s  see  the  diagram 


or       ,_     =  =_ 

w'      v  s  f 

The  negative  sign  is  used  merely  to  make  the  formula 
below  conform  in  looks  to  that  of  §  91  and  similar  ones. 
This  convention  has  no  effect  either  on  the  numerical 
value  of  F  (§  16)  or  on  the  sign  of  F. 

6 


THIN    LENSES  7 

If  the  lens  is  thin,  so  that  practically  w  =  w',  then,  by 
subtraction, 


./ '  r. 

r  =  radius  of  first  surface 

s  =  radius  of  second  surface 

/  =  focal  length  of  first  surface  for  incident  rays 

/'  =  negative  focal  radius  of  the  second  surface  (see 

preceding  paragraph)  =  — — - 
mfji  =  index  of  refraction 
16.   For  a  very  distant  source,  i.e.  u  =  oo,  we  get  the 


special  value  for  — 

J"-l*g+£)-fr-l)g 

1       1       JL 

v      u      F 


F  =  principal  focal 
length  of  the 
lens 

=  value  of  v  for 
(u  =  oo) 


That  is,  for  a  very  distant  object  (horizontal  rays)  all 
the  horizontal  rays  pass  through  F,  the  focal  point. 

17.    Since  distances  from  the  vertex  to  the  left  are  nega- 
tive, we  get 
For  a  double  convex  lens 

11  /      1       IV       1       11 


1  Heavy  black  face  type  indicates  numerical  values  without  regard 
to  direction;  light  letters  indicate  true  values,  taking  account  of  direc- 
tion where  this  is  necessary.  The  black  face  type  will  be  used  when 


8  THICK-LENS   OPTICS 

For  a  double  concave  lens 

11  n  (\    ,l\       1       1       1 

--^=(^-1)^  +  -^  =  -  =  --- 

18.  Starting  with  w  +  and  large,  if  F  is  +,  -  must  be 

greater  than  -,  in  order  to  make positive.     There- 
in                                   v       u 

fore  v  must  be  smaller  than  u,  and  the  image  is  nearer  the 
lens  than  the  object  is. 

If  r  >  s,  thus  making  the  lens  thicker  in  the  middle, 
or  if  r  <  0,  i.e.  negative,  thus  making  the  lens  a  double 

convex  lens,  then  -=  is  negative,  and  therefore  F  is  negative 
and  must  lie  on  the  left. 

Keeping  u  +  (or  on  the  right)  and  large,  which  makes  - 

vi 

small,  the  only  way  to  make <  0  (i.e.  neg.)  is  to 

1}  \Ji 

make  v  negative. 

In  other  words,  for  a  thinner-in-the-middle  lens,  later 
called  a  negative  lens,  u,  F,  and  v  have  the  same  sign 
when  the  object  is  real. 

19.  For  a  thicker-in-the-middle  lens,  later  called  a  posi- 
tive lens,  and  a  real  object,  /''and  v  must  have  different 
signs  from  u. 

Stated  in  another  way, 

For  a  +  lens,  the  further  focus  (from  the  object)  is  the 

active  focus. 

For  a  —  lens,  the  nearer  focus  is  the  active  focus. 

20.  Light  from  the  Right.  —  In  formulae  used  hereafter, 
the  positive  lens  (thicker  in  the  middle)  will  be  considered 

the  absence  of  direction  is  to  be  specially  emphasized.     In  other  cases 
the  context  will  indicate  whether  the  quantities  have  direction  or  not. 


THIN    LENSES  9 

as  having  a  negative  focal  length,  and  negative  lenses 
(thinner  in  the  middle)  as  having  a  positive  focal  length, 
because  the  effective  focus  lies  in  these  respective  directions. 

21.  Light  from  the  Left.  —  In  future  formulae,  the  posi- 
tive lens  will  be  considered  as  having  a  +  focal  length, 
and  the  negative  lens  as  having  a  —  focal  length,  because 
the  effective  focus  lies  in  these  directions  respectively. 

Since  this  makes  the  sign  of  the  lens  and  the  sign  of  the 
focal  length  concordant  (a  great  gain  in  uniformity),  the 
light  will  hereafter  be  assumed  to  come  from  the  left  unless 
otherwise  specified. 

Note.  —  The  diagram  in  §  15  was  so  taken  because  all 
the  quantities  are  positive,  thus  giving  a  normal  formula 
(see  §  10,  note)  applicable  to  any  diagram  when  we  take 
account  of  the  changes  in  sign  of  the  various  quantities. 
So  also  in  diagrams  of  §§  67,  72. 

22.  Notice  that  if  the  media  on  the  two  sides  of  the 
lens  are  not  the  same,  the  nearer  and  farther  focal-point 
distances  will  be  different  (Conf.  §  71).     This  will  easily 
be  seen  by,  in  the  preceding  investigation,  taking  /*/  for 
the  second  surface  instead  of  n,  and  finding  F  for  u  =  oo. 
Reversing  this  by  taking  p!  for  the  first  surface  and  p  for 
the  second  surface,  we  get  a  different  value  for  F. 

Unless  specifically  mentioned,  we  assume  that  we  have 
air  on  both  sides  of  the  lens,  the  usual  condition,  and  there- 
fore the  two  focal-point  distances  the  same. 

23.  Optical  Center.  —  The  investigation  of  §  65,  which 
may  be  read  here,  shows  that  for  any  lens  there  is  a  point, 
rays  passing  through  which  are  parallel  before  and  after 
refraction  by  the  lens.     For  a  thin  lens  this  point  must 
be  where  the  axis  of  the  lens  pierces  the  lens.     Rays  through 
this  point  are  not  changed  in  direction. 

24.  Diagrammatic    Investigations.  —  In    the    diagram- 


10 


THICK-LENS   OPTICS 


matic  investigations  which  follow,  the  determination  of 
image  from  object  will  be  made  by  means  of  two  definite 
kinds  of  rays. 

(a)  One  which  after  refraction  passes  through  the  proper 
focus  point,  i.e.  rays  from  the  object  parallel  to  the  axis 
of  the  lens,  as  if  from  a  distant  object, 

(6)  One  unchanged  in  direction  before  and  after  refrac- 
tion, i.e.  a  ray  through  the  center  of  the  lens  (or  in  the 
case  of  a  thick  lens,  the  nodal  points,  see  §  63). 

To  emphasize  the  characteristics,  say, 

Horizontal  rays  always  refract  to  the  focus. 

Central  (or  in  the  case  of  a  thick  lens,  nodal,  see  §  63) 
rays  pass  through  without  angular  deviation. 

Note.  —  Each  point  in  the  object  sends  rays  in  all  direc- 
tions, and  of  course  we  choose  those  which  serve  us  best. 

25.  Diagrammatic  Derivation  of  Image  (object  outside 
of  focal  distance). 


•f-lens 


JL     Horizontal  ray 


Object 

Jxis  of  lens 


Note  the  order  of  the  letters  (see  figure} ;  this  order,  formulated,  becomes 
a  rule  of  procedure. 

Convention.  —  The  object  is  represented  by  a  heavy' 
arrow,  the  image  by  a  light  arrow,  the  focus  by  F,  the 
thin  lens  by  a  vertical  heavy  straight  line. 

AB  represents  one  of  the  first  kind  of  rays,  which,  we 
know,  must  go  through  F;  AC  represents  the  second 


THIN    LENSES 


11 


kind,  which  goes  through  without  refraction.  Hence  D, 
the  point  where  the  two  rays  meet,  must  correspond  to  A, 
or  be  the  image  of  A. 

The  diagrammatic  procedure  of  finding  the  image  D, 
of  an  object  A,  may  be  symbolized  by  the  letters  (a  great 
help  in  future  diagrams) 

hi  — >  /2  to  c 
meaning : 

From  some  point  in  the  object  pass  horizontally  (h)  to 
the  lens  (I),  then  through  (— >)  the  right-hand  focus  (fa) 
to  the  center  line  (c).  The  intersection  will  be  the  image 
point,  (/i  would  mean  left-hand  focus.) 

Observe  that  F  and  v  are  +,  while  u  is  — .  Compare 
with  §  19. 

26.  In  a  similar  way  we  get  the  following  diagrams. 
When  u  >  F  numerically,  we  get  a  real  aerial  image 

which  can  be  made  visible  by  interposing  at  the  image 
point  a  piece  of  ground  glass.  The  rays  from  the  object 
to  the  lens  are  divergent  raySj  those  from  the  lens  to  the 
image  are  convergent  rays. 

27.  Virtual  Image  (object  inside  the  focal  distance). 


+  lens 


Object 

Object  inside  the  focal  distance:  the  dotted  lines  show  rays  made 
less  divergent  after  refraction. 

Diagram  showing  how  an  object  within  the  focal  point  gives  a  virtual 
image,  an  image  erect  instead  of  inverted  as  in  the  previous  diagram, 
and  which  cannot  be  made  visible  by  the  interposition  of  a  piece  of 
ground  glass.  Unlike  the  previous  case,  it  renders  the  divergent  rays 
less  divergent,  but  not  convergent. 


12 


THICK-LENS    OPTICS 


28.   Convergent  Rays. 

Notice  that  the  converging  rays  are  rendered  more  con- 
verging. 


lens 


Convergent  rays""- 


derial 
some 


object,  real 
previous  lens 


C  F 

Aerial  object  outside  or  inside  the  focal  distance. 
Diagram  showing  the  effect  of  interposing  a  +  lens  in  the  path  of 
converging  rays,  thus  producing  a  real  image. 

N.  B.  —  In  tracing  images,  notice  what  a  different  result 
we  get  for  the  same  position  of  the  object,  influenced  by 
its  being  a  real,  or  an  aerial  object  with  converging  rays. 
This  is  of  great  importance  in  tracing  images. 

29.    Negative  Lens.  —  In  the  same  manner  we  get  the 


-lens 


Divergent  rays:   virtual  image. 

following  progressive  diagrams,  showing  the  result  of  mov- 
ing the  object  outward  at  the  left  until  it  disappears  at  oo, 
coming  in  again  at  the  right  from  oo  and  moving  down  to 
within  the  focal  distance. 


THIN   LENSES 


13 


-lens 


Jl 


Object 
at* 


D 

YlTtUdl 

image  a  point 
Parallel  rays:  virtual  image  a  point 


-lens 
B    / 


Jirt 


inal\object 
<z  convergent 
^'~^       rays* 


Convergent  rays:   aerial  object  outside  the  focal  distance:   virtual 
image. 


derail          jieal  image 
-.object at  oo 5^ 


Fiftual'  •  -IT^C^-— x° 


.Convergent  rays:    aerial  object  at  focal  distance:   image  virtual  or 
real,  at  infinity. 

Note.  —  The  letters  A  BCD  have  the  same  progressive 
signification  as  in  §  25. 

Notice  that  the  formula  has  now  become  hl—>fi  to  c. 


14 


THICK-LENS   OPTICS 


30.   Collecting  these  results,  we  have  the  following  formu- 
lation of  procedure. 

Formula  for   diagram  tracing  of  rays  from   object  to 
image : 

Positive  lens        hi  — •>  /2  to  c 
Negative  lens       hi  — >  /i  to  c 

For  explanation  of  symbols,  see  §  25. 


Convergent  rays:   aerial  object  inside  focal  distance:   real  image. 

Notice,  what  has  been  elsewhere  spoken  of,  the  effective 
focus  for  a  -f  lens  is  the  right-hand  one;  for  a  —  lens, 
the  left-hand  one.  (Light  from  the  left.) 

31.  In  some  cases  it  becomes  necessary  to  trace  back 
the  rays  from  the  image  to  the  object  (see"§  37,  Ex.  13),  in 
which  case  we  have: 

Formulation  of  procedure  for  diagram  tracing  of  rays 
backward  from  image  to  object: 

Positive  lens        rf2l  \  \  c/2  to  c 
Negative  lens      rf\l  \\  cf\  to  c 

meaning,  draw  a  ray  (r)  through /2  to  the  lens  (I),  and  then 
along  a  parallel  to  a  center  line  through  /2  (||  c/2)  to  the 
center  line  (c).  f\  is  the  left-hand,  /2  the  right-hand  focus. 
The  use  of  this  formulation  will  be  found  to  be  of  great 
help  in  tracing  graphically  the  conjugate  points  of  a  lens. 
In  fact,  without  the  mechanical  aid  of  the  formulation, 


THIN    LENSES  15 

it  is  exceedingly  difficult  at  times  for  the  novice  to  do  so 
without  error,  especially  in  tracing  from  image  to  object. 

32.   Diagram  for  Oblique  Rays. 

By  §  25  A'  is  the  image  of  A.  A  second  ray  from  A, 
AB,  must  go  to  the 
same  point, "5  A'.  But 
by  §  3^  it  must  also 
go  through  the  point 
C,  where  the  parallel 
center  line  DC  pierces 
the  focal  plane. 

But   we    can    con- 
sider the  ray  A  B  as  an  oblique  ray,  and  the  formula  for 
oblique  rays  is  evidently 

rl  — ><#>2  1 1  to  c 

meaning,  draw  a  ray  (r)  to  the  lens  (I),  and  then  through 
(— 0,  the  secondary  focus  determined  by  a  parallel  through 
the  center  intersecting  with  the  focal  perpendicular  (<£2  ||), 
to  the  center  line  from  the  object  (c).  Omission  of  "to  c" 
gives  the  direction  of  the  refracted  ray,  independent  of  the 
origin. 

This  formula  evidently  includes  that  of  §  25  as  a  par- 
ticular case. 

The  reverse  formula  for  tracing  from  image  to  object  is 
rfal  ||  c<f>2  to  c,  the  interpretation  of  which  is  similar  to 
that  of  §  31,  of  which  this  is  the  general  case. 

For  a  negative  lens,  the  corresponding  formulae  are 

rl  — »  <£i  ||  to  c     and     r<£i  I  \\  c$\  to  c 

This  is  one  of  the  most  important  sections  in  the  book  for 
enabling  the  investigator  to  get  a  quick  and  graphic  idea 
of  the  location  of  the  images  due  to  a  succession  of  lenses, 


16 


THICK-LENS    OPTICS 


allowing,  as  it  does,  any  ray  to  be  traced,  whatever  the  effect 
of  the  lens  upon  it. 

Start  the  ray  from  the  intersection  of  the  object  with  the  axis. 
Each  new  intersection  with  the  axis  will  locate  an  image. 

The  same  principle  applies  to  the  refraction  through  a 
surface  and  reflection  from  a  surface,  the  surfaces  being 
typified  by  vertical  straight  lines,  as  in  the  case  of  lens 
surfaces. 

Since  the  point  A'  lies  on  the  line  A  A'  through  the 
optical  center,  its  position  will  not  be  changed  by  twisting 
the  lens  about  a  vertical  axis  through  the  optical  center. 
This  will  have  an  important  bearing  in  subsequent  sections. 

The  reverse  formulation  (see  §  31)  is  r<f>2l  \\  c<f>2  to  c,  and 
r<f>il  ||  c<£i  to  c,  for  +  and  —  lens  respectively. 

33.    As   an  example  of  the   application  of  the  section 


Diagram  illustrating  the  tracing  of  oblique  rays  from  the  object  A, 
through  the  three  lenses,  7,  //,  ///,  to  the  final  image,  ///.  The  foci 
and  images  of  the  lenses  are  marked  correspondingly  /,  II,  III.  The 
rays  in  their  different  courses  are  marked  by  encircled  numerals.  This 
is,  in  a  very  distorted  form,  the  course  through  a  compound  microscope, 
/  being  the  objective  and  //,  ///  the  lenses  of  the  eyepiece,  produ- 
cing the  virtual  image,  ///.  See  also  §  95,  Ex.  2. 


THIN    LENSES 


17 


above  and  §  25,  we  have  the  adjacent  diagram  of  rays 
passing  through  three  lenses:  I,  II,  III. 

Surface  refraction  can  be  traced  in  a  manner  similar  to 
that  of  §  32  by  the  formulae 

Incident  rays         rs  — ->  <fo  1 1 
Emergent  rays      rs  — >  <£2  1 1 

where  <fo  ||  means  the  point  determined  by  a  parallel 
through  the  center  of  curvature  and  a  perpendicular  (to 
the  axis)  through  the  focus  of  the  surface,  3  radii  (assuming 
/A  =  |,  §  12)  from  the  vertex,  measured  through  the  center. 
<£2  ||  similarly,  but  2  radii  (§  13)  from  the  vertex,  measured 
away  from  the  center.  (See  Appendix,  Figs.  1-4.) 

Example  1. — Try  this  method  in  checking  the  results 
of  §  72,  Exs.  7,  8,  10,  19. 

Example  2.  —  By  §  32,  drawing  a  ray  from  a  point  in 
the  axis  (see  (D  in  preceding  diagram),  show  that  in  order 
to  get  a  virtual  image  (the  case  of  a  positive  lens  used  as 
a  microscope)  the  object  must  be  within  the  focal  distance. 
(Conf.  §  93,  Ex.  4.) 

34.  Parallel  Rays  meet  in  the  Focal  Plane. 


By  Elementary  Geometry,  three  rays  through  a  point 
cut  off  proportional  parts  on  any  two  parallels;  hence 


18  THICK-LENS    OPTICS 

AD  =  OB      A'D'  OB         OG 

AE  =  OG  ~  A'E'  l7^'  ~  ~ArE' 

But  by  similar  triangles,  etc. 

OC_        OB         OG        OF^ 
CD'  ~  A'D'  ~  A'E'  ~  FE' 

Therefore  the  triangles  OCF  and  OD'E'  being  similar,  C 
and  F  are  equally  distant  from  the  line  OG 

That  is,  parallel  rays  focus  in  the  focal  plane  (the  plane 
through  the  focus  perpendicular  to  the  axis)  at  a  point 
determined  by  the  center  line;  and,  conversely,  rays  from 
a  point  in  the  focal  plane  emerge  parallel,  parallel  to  a 
line  from  the  point  through  the  center. 

35.   Standard  Formula.  —  One  formula  (viz. =  - 

V         v      u      / 

is  used  throughout  the  book,  the  proper  sign  (+  or  — ) 
being  given  to  the  numerical  values  when  used.  The  use 
of  the  two  formulae  of  §  17,  one  for  the  positive  lens  and 
one  for  the  negative  lens,  as  is  the  practice  of  some  writers, 
is  apt  to  lead  to  confusion,  since  both  formulae  apply  to 
both  lenses  under  some  conditions.  It  is  the  difficulty  of 
distinguishing  these  conditions  that  makes  the  trouble  for 
the  non-expert.  Hence  the  decision  at  the  head  of  this 
section,  since  then  the  only  difficulty  arises  from  the 
selection  of  the  +  and  —  signs.  This  selection  is  guided 
by  the  rules  of  the  next  two  sections.  (See  note  to  §  17.) 


USE  OF  THE  FORMULA       -       = 

(See  diagrams  of  §§  25-29.) 

36.   Positive  Lens.  —  Real  object,  diverging  rays,  real 
image  (object  outside  of  F): 


THIN    LENSES  19 

/  and  v  must  have  a  different  sign  from  u. 
Real  object,  virtual  image  (object  inside  of  F): 

u  and  v  must  have  a  different  sign  from  /. 
Converging  rays,  aerial  object: 

u,  v,  and  /  have  the  same  sign. 

For  light  coming  from  the  left,  /  is  positive  (for  light  from 
the  right,  /  is  negative).  (See  §  76.) 

37.   Negative  Lens.  —  Real  object,  diverging  rays: 

u,  v,  and  /  have  the  same  sign. 
Converging  rays  (converging  outside  of  F),  virtual  image: 

u  different  sign  from  v  and  /. 
Converging  rays  (converging  inside  of  F)  real  image: 

u  and  v  have  different  sign  from  /. 

For  light  coming  from  the  left  /  is  negative  (for  light  from 
the  right,  /  is  positive).  (See  §  76.) 

EXAMPLES 

Check  each  calculation  by  an  actual  drawing,  to  scale 
(§  38),  to  avoid  large  errors;  guide  the  drawing  by  the 
formulation  of  §§  30,  32,  and  see  §§  38-41. 

Decide  on  the  direction  of  the  ray,  thus  fixing  the  sign 
of  u  and  /  (say  from  the  left).  If  from  the  left,  /  will  be 
positive  for  a  positive  lens  and  negative  for  a  negative  lens; 
u  will  be  negative  for  a  real  object  or  an  aerial  object  with 
diverging  rays  therefrom,  and  positive  for  an  aerial  object 
and  converging  rays.  (See  §  76.) 

From  the  data  given  find  the  other  elements. 

1.    Positive  lens  with  F  =  1  ft. 

(a)  u  =  -  11  in.  /.  v  =  -  11  ft. 

(6)  u  =  --  10  in.  /.  v  =  -  5  ft. 

(c)  u  =  -  1  in.  /.  v  =  -  TV  ft. 

(d)  u  =  -  20  ft.  /.  v  =  f  $  ft. 


20  THICK-LENS    OPTICS 

(e)  u  =  -  2  ft.  /.  v  =  2  ft. 

(/)  u  =  -  li  ft.  /.  v  =  3  ft. 

2.  u  =  -  2f  .'.  v  =  2f 

3.  u  =  -  6,  v  =  1  .'.  F  =  f 

4.  w  =  3  in.,  i;  =  18  in.        .'.  F  =  -  3f  in. 

5.  u  =  12,  v  =  1  /.  F  =  H 

6.  r  =  5,  s  =  —  7,  negative  lens,  /*  =  $,  w  =  60. 
Ans.  F  =  \5,  v  =  -V/,  double  concave. 

7.  Positive  lens,  r  =  7,  s  =  5,  /*  =  |,  w  =  60. 

^.ns.     Light  from  right,  F  =  —  35,  v  =  —  84,  concavo- 
convex. 

8.  Positive  lens,  r  =  --  7,  s  =  5,  /*  =  f ,  w  =  60. 

^.ns.     F  =  —  ™->  v  =  —  f  f,  double  convex,  light  from 
right. 

9.  Negative  lens,  r  =  5,  s  =  7,  n  =  |. 
Ans.    /  =  15,  /'  =  -  21,  F  =  35. 

10.  Negative  lens,  r  =  7,  s  =  5,  /*  ==  |. 

Ans.     Light  from  left,  /  =  21,  /'  =  --  15,  F  =  -  35. 

11.  Negative  lens,  r  =  —  7,  s  =  —  5,  /A  =  |. 

Ans.     Light  from  right,  /  =  -  21,  /'  =  15,  F  =  35. 

12.  Convex  lens,   light   from   right,   F  =  5.813,   object 
30.56  in  front.     Where  is  the  image? 

Ans-     -  =  57T^+      AM  o=0-03273 -0.1720=  -0.1392 
v     60. ob      —  o.olo 

«  -     /.  »  =  -  7.183  to  left. 


THIN    LENSES  21 

13.  A  telescope  has  a  field  glass  of  23f  inches  focus  and 
an  erecting  eyepiece  composed  of  4  lenses  as  follows, 
reading  towards  the  eyepiece,  2,  If,  If,  If  inch  focus, 
with  the  separations  2J,  4,  2  inches.  To  trace  the  con- 
jugate foci. 

Since  the  last  four  lenses  are  fixed  and  the  focussing  is  done 
by  adjusting  the  combination  relative  to  the  field  glass, 
we  take  as  the  starting  point  the  virtual  image  seen  by 
the  eye.  This  will  be  seen  at  a  distance  determined  by 
the  "set  of  the  eye"  of  the  observer.  (See  §  109.) 

We  assume  the  "far  set"  eye  and  the  rays  to  the  virtual 
image  parallel.  This  makes  the  object  for  the  fourth  lens 
(counting  from  the  left)  at  the  focus  of  that  lens;  and 
indicating  by  v\,  u\  the  conjugate  distances  for  the  first 
lens,  etc.,  we  have  the  following  series  of  conjugate  distances. 

11 

V.i   =    00  ,  U4    =      -  -- 


,3  =  2  -        =  .625.  --•   .•-«,-.  987 


„  =  4  +  .937  -  4.937.   ^  -  I  =  A.    ,.  %  =  -  3.012 

*=-  3.012  +  2.25  =-  .762.     -^  -  1  -  J 

.'.  ui  =  -  .552 

This  shows  that  the  focus  of  the  field  glass  should  be 
.552  inches  in  front  of  the  first  lens.  This  is  approximate 
only,  since  the  real  lenses  must  be  treated  as  thick  lenses, 
as  in  Chapter  III. 


22 


THICK-LENS   OPTICS 


The  lenses,  with  their  corresponding  images  and  foci,  are  designated 
by  Roman  numerals.  The  dotted  lines  show  the  course  of  a  ray  from 
the  foot  of  the  aerial  object,  with  the  construction  lines  .(§  35).  Notice 
how  the  inverted  object  (aerial)  is  converted  into  an  erect  one  by  the 
second  lens.  The  virtual  image  being  at  infinity,  the  last  course  of 
the  ray  is  horizontal,  as  shown. 


GRAPHIC  CHECK  ON  CALCULATIONS 

38.  Inspection  of  the  diagrammatic  constructions  will 
show  that  they  fall  under  one  or  the  other  of  the  following 
diagrams,  or  modifications  of  these. 

Where  all  the  quan- 
tities are  on  one  side  of 
the  zero  line  (lens  line), 
as  in  the  left-hand  dia- 
gram, we  have  (by  sim- 
ilar triangles) : 

a          v        b          v     ,  v    .  v      a  +  b       1 

— 7  =  -i  — r~r  =  «  whence  — r  -f  =       .   r  =  * 
a  +  b      u'  a+b      f  u      f      a  +  b 

1    ,1       1 

or  — r  7  =  - 

u      f       v 

which  is  the  same  as  in  §  16. 

Worded,  this  becomes,  The  reciprocal  of  the  mid  line  = 
the  sum  of  the  reciprocals  of  the  end  lines  (when  all  the 
quantities  are  on  the  same  side  of  the  zero  line). 


THIN   LENSES  23 

39.  Similarly  for  the  right-hand  diagram,  unless  we  take 
into  account  the  signs  of  the  quantities,  in  which  case: 

Where  two  of  the  quantities  are  on  different  sides  of 
the  zero  (lens)  line  (right-hand  diagram)  The  reciprocal 
of  the  mid  line  =  the  reciprocal  of  the  end  line  on  the  same 
side  of  the  zero  line  as  the  mid  line  —  the  reciprocal  of  the 
end  line  on  the  other  side  of  the  zero  line. 

USE  OF  THE  DIAGRAMS.  —  To  insure  accuracy  in  signs  and 
to  detect  material  (large)  errors,  plot  these  diagrams  to  scale. 
Lay  off  the  end  lines  any  distance  apart,  draw  the  diagonals 
and  see  if  the  mid  line  fits  in  size  and  sign.  Or  lay  off  f  and 
u  (any  distance  apart)  and  then  by  means  of  the  two  diagonals 
determine  v. 

EXAMPLES 

1  1  J_  1  1  1 

-11"    =TTl2      12'  -10"  -5-12~l2' 


1=*-  J-  Ex  6    -i-  =  U      l 

61-6  18     3  "*         18 

7  '   5 


GRAPHIC  CHECK  ON  CALCULATIONS 

This  method  is  given  in  some  detail  because  so  many  books  use 
one  or  the  other  of  the  diagrams. 

40.   Pos.  lens  with  +  f.     By  similar  triangles 
u      u  +  v'  111 


24 


THICK-LENS    OPTICS 


which  is  the  same  as  the  equation  of  §  17  for  a  positive 
lens. 

USE  OF  THE  DIAGRAM.  —  Set  off  /  and  u  in  proper  size 
and  direction,  draw  the  line  through 
their  ends:  its  intersection  with  the 
45°  line  will  give  v  both  in  size  and 
sense,  u  on  the  right  indicates  an 
aerial  image  made  by  some  preceding 
0  lens. 

Variations  of  this  are,  as  the  object 

moves  from  the  left,  being  aerial  when  on  the  right  of 
the  lens,  the  rays  of  light  coming  from  the  left, 


/ 


45' 'line 


© 


Diagrams  showing  the  relative  sizes  and  positions 
of  u  and  v. 

®  Real  object  beyond  the  focal  distance,  real 
image,  inverted.  (Light  from  the  left.) 

(D  Real  object  within  the  focal  distance,  virtual 
image,  erect. 

(3)  Aerial  object  within  the  focal  distance,  real 
image,  erect. 

@  Aerial  object  beyond  the  focal  distance,  real 
image,  erect. 

41.   Negative  lens  with  —  f.     In  th6  same  manner,  we 
get  the  following  diagrams: 

USE  OF  THE  DIAGRAMS.     (See  §  40.) 


THIN    LENSES 


25 


Notice.  —  When  u  and  v  have  the  same  sign,  the  image  is 
erect.  When  u  and  v  have  a  different  sign,  the  image  is 
inverted.  (Both  lenses.) 


0  Real  object  beyond  the  local  distance,  virtual  image,  erect. 

0  Real  object  within  the  focal  distance,  virtual  image,  erect. 

®  Aerial  object  within  the  focal  distance,  real  erect  image. 

0  Aerial  object  beyond  the  focal  distance,  virtual  inverted  image. 

POWERS  OF  LENSES:    DIOPTERS 


42.    In  the  expression  --- 
v       u 


-.•>  -f  is  called  the  power 
f   f 


of  the  lens  (to  alter  a  light- wave  front),  and  when  v  and  u 
are  expressed  in  meters  (or  its  practical  equivalent,  40 
inches),  the  power  units  are  called  diopters. 

Similarly,  -  =  p  and  -  =  p'  are  called,  for  convenience 

v  u 

of  reference,  powers  of  the  distances. 

If  we  call  the  power  of  the  lens  Z>,  then 


D  =  P  - 


-  1)  (q  -  q') 


Dioptric  units  are  generally  used  by  opticians  in  con- 
nection with  (thin)  spectacle  lenses.  The  power  of  a 
combination  of  lenses  equals  the  sum  of  the  powers.  (See 
§75.) 


THICK-LENS    OPTICS 


EXAMPLES 

43.    Ex.  1.     u  =  -  80  cm.,  /  =  20  cm.,  v  =?     (Light 
from  left.) 


Ans.    p  =     = 

100 
•'•  *  "  376  "  26f 


Ex.  2.     u  =  -  20  in.,  ti  =  5  in.,  /  =? 


Ex.  3.    /  =  10,  u  =  -  8,  v  =  ?     (Positive  lens.) 

n       40  40 

Ans.     D  =  JQ  =  p  -  -—g  =  4  =  p  +  5.     .'.  p  =  -  1. 

Ex.  4.  Four  lenses  in  contact:  (a)  a  plane  concave  of 
4  diopters;  (b)  a  positive  meniscus  of  r  =  2  in.,  s  =  5  in.; 
(c)  a  biconvex  of  50  cm.  focus;  (d)  a  biconcave  of  33  J  cm. 
focus.  What  is  the  focus  of  the  combination?  (Light 
from  left.) 


Ans.     (a)  =  -  4  D.     (b)  =      ~  -  •         =  6  D. 

W-E-.ix   W-'-g  —  ift 

Therefore,  combination  =  -4D  +  6D  +  2D-3D  =  Z>, 
and  the  result  is  a  positive  lens  of  100  cm.  focus,  projecting 
a  real  inverted  image.  (See  §  25.) 

44.  The  focal  length  expressed  in  inches  gives  the  num- 
ber of  the  lens.     (Obsolete.) 

45.  Spectacles  for  Farsighted.  —  Positive  lens,  virtual 
image. 


THIN   LENSES  27 

The  formula  of  §  16  becomes: 

1  1 

-v        -10 


the  10  and  v  being  taken  negative,  because  we  want  a 
virtual  image.  See  diagram  of  §  27.  v  must  be  the  nearest 
'distance  at  which  the  wearer  can  conveniently  see  without 
spectacles,  10  being  the  distance  at  which  he  holds  the 
book.  The  image  is  virtual. 

46.   Spectacles  for  Nearsighted.  —  Negative  lens. 

In  this  case  the  formula  of  §  16  becomes: 

1  1  1 

-  v       —  u       -  f 

u,  v,  and  /  have  the  same  sign,  §  37.  v  must  be  the  greatest 
distance  at  which  the  wearer  can  see  clearly  without  spec- 
tacles, 10  inches  being  the  distance  at  which  the  book  is 
held.  The  image  is  virtual.  If  /  is  less  than  v,  he  can 
see  objects  at  all  distances  over  10  inches,  since  the  virtual 
image  is  always  within  his  visual  distance.  See  first  dia- 
gram of  §  29. 

EXAMPLES 

1.  If  longest  distance  for  distinct  vision  is  15  cm.,  what 
lens  will   enable   the   wearer   to   see   all   distant   objects? 
Ans.     15  cm.  or  under. 

2.  Book  is  held  at  1  ft.  with  concave  6  in.  focal  lens. 
Where  is  the  image?     Ans.    4  in. 

3.  A  man  can  read  distinctly  at  15  cm.     What  lens  must 
he  use  if  he  wants  to  read  easily  at  60cm.?     Ans.    f  =  —  20. 

4.  If  the  nearest  distance  for  distinct  vision  is  15  inches, 
what  focal  length  of  spectacle  is  required  if  the  book  is 
held  at  10  inches?     Ans.     30  in. 


28 


THICK-LENS   OPTICS 


5.    If  the  shortest  distance  for  distinct  vision  is  1  m., 
what  length  spectacle  is  wanted  for  object  at  25   cm.? 


Ans.     -7  =  -7^  —  1=4  —  1=3  diopters,  or/  =  33 J  cm. 


.25 


6.  Best  vision  at  3  ft.     With  1  ft.  spectacles  the  book 
should  be  held  at  .  .  .?     Ans.     9  in. 

7.  A  longsighted  person  with  +  glasses  of  40  cm.  length 
finds  he  must  hold  the  book  no  nearer  than  30  cm.  for 
comfort.     What  is   his  nearest  point  of  distinct  vision? 
Ans.     120  cm. 

8.  A  longsighted  person  can  only  see  distinctly  at  48 
cm.  or  more.     By  how  much  will  he  increase  his  range  of 
vision    with    convex   spectacles    of   32    cm.    focus?     Ans. 
48  -  19.2  =  28.8. 

9.  A  person  whose  distance  of  most  distinct  vision  is 
20  cm.,  uses  a  reading  glass  of  5  cm.  focus.     How  far  from 
the  book  must  it  be  held?     Ans.     4  cm. 

47.   Magnification  for  Convex  Lens,  Real  Image  (Cam- 
era, etc.). 


•f-lens 


a.         Ill  1,11.  ...       . 

Since =7,  or  -  H —  =  j  for  positive  lenses, 

v       u      f         v       u       f 

y 

-  =  M  =  magnification  factor  =  ratio  of  image  to  object 


THIN    LENSES  29 

=  ratio  of  image  distance  to  object  distance 
v  f 

=  f  "      =  iTH 

T.  _       .„     ,.  size  of  image       image       b      v  —  f 

Magnification  = ~ — r^ — -  =  -       --  =  -  =  — 7 — 

size  ot  object  c  a 

u  =  ^  -{-  f  =  distance  to  object 

v  =  f M  +  f  =  distance  to  image 
If  M  =  1,  then  u  =  2/,  v  =  2/. 

If  M  =  —  1,  which  is  equivalent  to  saying,  image  same 
size  but  not  inverted,  then  u  =  0,  v  =  0. 

Note.  —  Ordinarily  (when  we  are  taking  account  of  direc- 
tion) -f-  M  indicates  erect  image;  —  M,  inverted  image. 
Do  not  get  the  two  cases  confused. 


EXAMPLES 
1.   In    Ex.    9,    §  46,    what    is    the    magnifying    power? 


Ans. 


—  -  T 
5  —  4 


2.  An  engraver  uses  a  4  in.  focus  magnifying  glass,  close 
to  the  eye.     Where  must  he  hold  his  work  to  get  a  magnifi- 

4 

cation  of  4?     Ans.     -  -  =  4.         .*.  u  =  3. 
4  —  u 

3.  An  object  is  3  ft.  in  front  of  a  6  in.  lens.     What  is 
the  magnification?     Ans.     J/(3  —  J)  =  i- 

4.  v  =  8,    4-inch  lens.     Ans.     M  =  1,  u  =  8. 

5.  v  =  12,  4-inch  lens.     Ans.     M  =  2,  u  =  6. 

6.  v  =  16,  4-inch  lens.     Ans.     M  =  3,  u  =  5J. 


30  THICK-LENS    OPTICS 

7.  v  =  20,  4-inch  lens.     Ans.     M  =  4,  u  =  5. 

8.  An  object  8  cm.  high  is   1   m.  from  an  equiconvex 
lens  of  index  of  refraction  1.5  and  radius  of  curv.  0.4  m. 
Where  is  the  image  and  what  is  its  size?     Ans.     Image 
f  m.  on  other  side  and  of  height  5J  cm. 

9.  In  Ex.  1,  §  37,  what  are  the  magnifications?     Ans. 
12,  6,  TV,  A,  1,  2. 

10.  Converging  lens,  with  object  5  in.  from  lens.     Image 
=  6  times  the  object.     Where  is  the  image  and  what  is 
the  focal  length?     Ans.     v  =  30  in.    /  =  -\°-. 

11.  Required  an  image  of  3  mag.  by  lens  of  F  focal 
length.     How  far  must  the  screen  be  from  the  object? 

Ans.    \F.  1  +  -L  =  1.          .-.x  =  \F. 

6  x       6  x       r  6 

12.  An  object  is  a  distance  d  from  a  screen,  and  a  thin 
pos.  lens  is  placed  to  form  an  image.     If  the  lens  be  moved 
a  distance  p  =  Vd2  —  4  df,  another  image  will  be  formed 
whose  linear  dimensions  are  to  those  of  the  former  as 
(d  -  p)2  to  (d  +  p)\ 

13.  A  disc  1  inch  in  diameter,  8  inches  from  the  eye,  is 
seen  through  a  convex  lens  of  8  inches  focus,  placed  half^ 
way  between.     What  should  be  the  diameter  of  the  lens 
to  see  the  whole  of  the  disc  at  once?     What  is  the  distance 
of  the  image  from  the  eye?     Ans.     Diam.  of  lens  =  f  in. 
Dist.  of  image  from  eye  =  12  in. 

14.  A  concave  lens  is  blackened  except  a  circle  of  4  cm. 
diameter  at  the  center.     A  beam  of  sunlight  through  this 
gave  an  illuminated  circle  of  20  cm.  diameter  on  a  screen 
64  cm.  from  the  lens.     Show  that  the  focus  of  the  lens  is 
—  16  cm.     Use  the  first  diagram  of  §  29. 

15.  If  u  =  2/,  then  v  =  2/,  M  =  1. 


THIN    LENSES  31 

j|/f  f         u  —  f'       f 

16.    If  u  is  very  large,  show  that  -=-^7  =  -    — ;  •        ,      =  J—, 

M         u  —  j        j  j 

practically,  and  therefore  that  for  distant  objects  the  sizes 
of  the  image  are  proportional  to  the  focal  radii  of  the  lenses. 
48.   Copying,  Enlarging,  etc.,  with  the  Camera. 
To  find  distance  to  plate,  etc.,  for  given  magnification 
or  reduction. 

v  =  distance  from  lens  to  screen  (plate) 
=  camera  extension. 

=  F  (M  +  1)  =  ^  +  F     (§47) 
M  =  magnification  factor 
N  =  reduction  factor;  M  =  -~ 
F    =  focal  length  of  the  lens 
u  =  distance  from  lens  to  object 

=  F  +  ^    =  F  +  NF 

Strictly  these  distances  should  be  measured  from  the 
nodal  points  (see  §  63),  but  approximate  values  and  meas- 
urements are  sufficient  for  a  first  adjustment,  the  final 
being  made  by  trial. 

EXAMPLES 

1.  6  in.  lens,  12  in.  drawing,  4  in.  copy,  whence  N  =  3. 

2.  6  in.  lens,  4  in.  plate,  12  in.  copy,  whence  M  =  3. 
Ans.     v  =  24,  u  =  8. 

3.  A  candle  stands  a  yard  from  the  screen.     What  lens 
and  where  must  be  used  to  get  an  image  5  times  as  large? 

Ans.     5  in.  lens,  30  in.  from  the  screen. 


32  THICK-LENS    OPTICS 

49.   Best  Position  for  Condenser. 


Condenser  of  joe  al  length  f 


Jfounting  of 
the  objective 


JTejative 


\_ 
V 

1 

v 


-  =  M  =  Magnification 

v  =  (1  +  M)  f 
1  +  M 


u  = 


M 


f 


[§16 


[§48 


To  get  all  the  picture  within  the  cone  of  rays, 
a  V 

g~~  V  -x 

whence  x  can  be  calculated. 

To  get  the  cone  of  light  to  just  fill  the  mounting, 

a  V 

c  =:  V  -  x  -  u  +  b 

whence  u  can  be  found. 

For  sunlight,  V  =  F,  (U  =  oo),  whence 
a 


F  = 


g  - 


THIN    LENSES  33 

If  x  =  °'       =          •  -      whence  u 


can  be  found. 

Example.  —  F  =  3,  /  =  6,  M  =  5,  a  =  f  ,  c  =  },  b  =  f  . 

t^i.     J-          -    C7-317 


50.   Exposure. 

T  =  time  of  exposure  for  distance  v  of  image,  with  aper- 
ture d 


(the  subscripts  indicating  the  corresponding  quantities  for 
some  known  exposure  with  a  satisfactory  result) 


tZY  (^ .  P\.  Tl  =  (M_±_T)2 

+  1\2  U.S. 


/Y 

/J    ' 


+  1J   U.S., 


U.S.  =  U.S.  numbers 
/,  /i  =  the  /  numbers;  see  §  51 
M,  MI  =  magnification 

In  use,  disregard  the  letters  in  which  there  has  been  no 
change  of  conditions,  and  see  note  at  the  end  of  Example  1. 

EXAMPLES 

1.  With  an  8  in.  lens,  with  //20  enlarging  5  times,  40 
seconds  exposure  was  required.  What  exposure  is  required 
for  a  9  in.  lens  //30  and  enlarging  6  times? 


sec. 


34  THICK-LENS    OPTICS 

Notice  that  the  /  number  determines  the  exposure  with- 
out regard  to  the  focal  length.  E.g.  //20  requires  the  same 
time  whatever  the  lens. 

_.,  „  focal  length 

ol.    slowness  tactor  = — rr-2 =  n,  written  = 

aperture  diameter 

f/n  and  called  the  /  number. 

Time  of  exposure  varies  as  the  square  of  the  slowness 

factor.     For  example,  a  ~-  iens  requires  -^j-  the  exposure 

/ .  169       132 

of  a  -  lens.  ^    =  ^ 

The  U.S.  numbers  give  the  relative  time  of  exposure, 
+  -UJ1.U 

whence  we  can  find  corresponding  /  and  U.S.  numbers  by 
the  formulae 


/  number  =  4  VU.S.  number 
(j  numberV 


U.S.  number  =  , 

52.  Hyperfocal  Distance.  —  When  the  object  is  too  near 
to  the  lens,  what  produces  confusion  in  the  picture  is  the 
overlapping  of  images.  It  is  found  that  a  slight  overlap- 
ping is  not  distinguishable.  This  occurs  when  the  two 
images  of  a  given  point  are  not  more  than  T^  inch  apart. 
Hence  up  to  the  point  in  front  of  the  lens  where  the  separa- 
tion of  images  is  not  more  than  T£<j  inch,  all  objects  will 
apparently  be  in  focus. 

The  distance  of  this  point  from  the  lens  is  called  the 


THIN   LENSES 


35 


hyperfocal  distance,   the   distance  to  the  nearest  distinct 

object     beyond 

which    all    objects 

are    apparently  in 

focus. 


The  different    P. 
images  of  a  given 
object  areevidently 
scattered  over  a  circle.     The  largest  circle  permissible  with- 
out confusion  is  called  the  circle  of  confusion. 

F  =  focal  length  in  inches 
P  =  nearest  distinct  object 
Q  =  image  of  P 

a   =  radius  of  circle  of  confusion 
d  =  diam.  of  stop  in  inches 


f  =  -7  =/ number 


1 
200 


m. 


By  sim.  triangles 
But  §  17 


d/2 


l_  _  __  _ 

F  +  x~*~  u  ~  F 


1 


u  =  F 


F  +  x       F  (F  +  x) 


Fd 
2a 


focal  length  X  diam.  of  stop 
.  diam.  of  circle  of  confusion 


/Till 

100  F2  . 


100  F2 


36 


THICK-LENS    OPTICS 


=  distance  to  nearest  distinct  object, 
beyond  which  all  objects  are 
apparently  in  focus 

Note.  —  The  diagram  is  drawn  for  a  thick  lens.  If  the 
two  vertical  dotted  lines  should  be  brought  together,  it 
would  make  the  proper  diagram  for  a  thin  lens,  with  no 
change  in  the  mathematics. 

53.     Magnification  and  Reduction  for  Negative  Lens. 
"Lens 


Objec\ 


u 


T->    i     ,-  1         image       f  —  v  v 

Reduction  =  —  =  -T^T  =  —7—  =  1  -  ? 

N       object          /  / 

u  f 

f+u      f+u 

As  the  object  is  moved  nearer  the  lens,  the  image  grows 

larger,  until  with  the  object  at 
the  lens  we  get  unit  reduction. 
54.  If  converging  rays  (due 
to  convex  lens)  are  coming  from 
the  left,  we  have  the  following 
diagram,  the  real  image  of  the 
convex  lens  being  the  aerial 
object  of  the  negative  lens. 


ens 


Image 


Serial 
object 


Image 
Object 


magnification  of  image  of  the  +  lens,  due  to 
the  —  lens. 


aerial  object      f  —  u 


THIN   LENSES 


37 


This  is  the  telephoto  combination  spoken  of  in  Chapter  V. 
v  is  there  called  the  bellows  extension  and  denoted  by  E, 

p< 
whence         M  =  1  +  —     and     E  =  f  (M  -  1) 

55.    Magnifying  Power  of  a  Positive  Lens  used  as  a 
Microscope. 

+lens 


In  this  case  the  image  seen  by  the  eye  is  virtual  and 
subtends  the  angle  0  shown  in  the  diagram.  Since  this 
angle  is  very  small,  it  is  measured  by  its  tangent,  and 

0  =  tan  9  =  **- 

The  angle  subtended  by  the  object  is 

c 


u  +  y 


and  the  ratio  of  these  is  the 


Apparent  magnification  =  — 

9 


a 


f  —  u      cd 

=  f  (v  +  y)  +  vy 
fd 


cd 

?          / 
f-u 

fv 

t  +  7 

i_      i 

—  v       —  u 


38  THICK-LENS  OPTICS 

,  (<*-y)y 
id 

This  is  a  maximum  =  1  +  —j  when  y  =  -  d,  which  ex- 
plains why  some  readers  like  to  push  their  spectacles  down 
towards  the  end  of  the  nose. 

The  distance  d,  the  distance  for  distinct  vision,  is  gen- 
erally taken  as  10  inches  (25  cm.),  but  should  be  taken  a 
specific  value  for  each  observer. 

56.  Ordinarily  the  conventional  magnification  of  the 
lens  is  stated  as  the  ratio  of  the  actual  size  of  the  image  to 
that  of  the  object,  viz. 

a       v 
Conventional  magnification  =  -  =  - 

v  (v  +  f ) 


vf 


EXAMPLES 

1.  A  convex  lens,  of  focal  length  £  inch,  is  used  by  a 
14  inch  (nearest  distance  for  distinct  vision)  eye.     What  is 
the  magnification? 

Ans.     Mag.  =  14/t  +  1  =71. 

2.  A  2  yard  eye  uses  a  2  ft.  lens.     How  far  from  the 
glass  should  the  object  be  placed? 

Ill  -  3,, 

Ans.     -z  --  =  —  o5  therefore  u  =  =  ft. 

D          U          —   A  & 

3.  Check  Ex.  2  by  comparison  of  images,  graphic  con- 
struction, §  32. 


CHAPTER   III 

THICK  LENSES 

57.  One  method  of  finding  the  equivalent  focus  of  a 
thick  lens  is  to  select  a  thin  lens  (spectacle  lens)  which 
will  give  on  the  ground  glass  an  image  of  exactly  the  same 
size  as  the  thick  lens  gives,  the  object  being  very  distant. 
The  focal  length  of  this  thin  lens  will  be  the  focal  length 
of  the  lens  under  consideration.     On  the  mounting  of  the 
thick  lens  mark  off  this  distance  from  the  ground  glass 
when  in  focus.     This  point  is   called  the  principal  point 
of  emergence. 

Turning  the  lens  around,  we  get  a  similar  point  for  the 
other  end  of  the  lens.  These  two  separated  points  mark 
the  points  from  which  evidently  measurements  for  focal 
radii  are  to  be  made,  and  correspond  to  the  optical  center 
of  a  thin  lens. 

Like  optical  centers,  these  principal  points  will  be  found 
to  be  points  around  which  the  lens  can  be  twisted  (about 
a  vertical  axis)  without  affecting  the  image  on  the  ground 
glass. 

58.  Suppose  an  object  to  take  the  successive  positions 
a,  6,  c,  and  then  the  aerial  objects  at  d  and  e,  with  the 
resulting  images  a',  6'  ... 

39 


40 


THICK-LENS    OPTICS 


At  the  optical  center  of  the  thin  lens,  and  only  there 
(Conf.  §  47),  will  the  object  and  image  have  the  same  size 
and  sense  (image  not  inverted). 


Diagram  showing  the  images  resulting  from  successive  positions 
of  the  object,  and  the  resulting  changes  in  size  of  the  image; 
i.e.  its  magnification  with  the  corresponding  images  a',  &',  c'.  .  . 

59.  In   §  32  we  found  that  revolution  about  an  axis 
through  the  optical  center,  the  point  from  which  the  focal 
radius  is  measured,  did  not  disturb  the  image.     Experiment 
shows  that  revolution  about  an  axis  through  the  principal 
point  of  emergence,  the  point  from  which  we  measure  the 
focal  distance,  does  not  disturb  the  image  of  a  distant 
object.     This  is  the  point  around  which  panoramic  cameras 
are  revolved. 

60.  Principal  Points.  —  Just  as  image  and  object  have 
the  same  size  and  relation  at  the  optical  center  of  a  thin 
lens,  so  we  might   anticipate  that  for   a   thick   lens  the 
image  and  object  would  have  the  same  size  and  relation  at 
the  principal  points,  the  points  from  which  the  radii  are 
measured.     (Conf.  §  69.) 

61.  We  can  find  these  principal  points  as  follows: 


Horizon tal  ray 

«   / 


THICK   LENSES 

/*~**\ 

/         ^Principal  plane 


41 


Principal  pl 


Diagram  showing  construction  for  the  determination  of  the  principal 
points.  The  order  of  the  letters  indicates  the  order  of  construction. 
6,  c,  found  by  §  7.  The  upper  half  gives  the  construction  for  one 
principal  point,  the  lower  half  for  the  other. 

Caution.  —  This  construction  applies  strictly  only  to 
points  near  the  axis,  but  it  serves  to  illustrate  the  principle 
for  future  use. 

Theoretically  we  could,  by  picturing  the  surfaces  as 
straight  lines,  get  a  correct  graphic  construction,  but  the 
disparity  between  the  thickness  of  the  lens  anji  the  radii 
is  generally  so  great  that  the  graphic  construction  is  of 
little  value  by  reason  of  its  acute  intersections. 

62.    Since   there  are   two  points  around  which  we  can 


Diagram  illustrating  the  apparent  horizontal  transference  between 
the  principal  planes. 


42 


THICK-LENS    OPTICS 


revolve  the  lens  without  effect  on  the  image  (the  lens 
being  reversed  so  as  to  make  each  one  a  point  of  emergence), 
i.e.  two  points  like  the  optical  center  of  the  thin  lens,  two 
points  where  the  object  and  image  have  the  same  size  and 
relation,  as  shown  by  the  diagram  (Conf.  the  diagram 
of  §  61),  the  effect  is  as  if  the  rays  from  the  object  passed 
to  the  first  principal  plane  and  then  were  transferred  hori- 
zontally to  the  other  principal  plane  so  as  to  keep  the 
object  and  image  the  same  size  from  plane  to  plane.  (Conf. 
§  105  after  reading  §  64.) 

This  equivalent  pair  of  parallel  surfaces  is  called  the 
equivalent  thin-split. 

63.  Nodal  Points.  —  The  following  construction  gives 
two  new  points  of  importance  called  nodal  points. 


Font* 


~Foc 


Diagram    illustrating   the    location   of    the   nodal 
points. 

From  A,  a  point  in  one  focal  plane,  draw  the  horizontal 
ray  A  B,  which  is  of  course  refracted  to  the  focus  Fr.  Draw 
AC  parallel  to  BF'.  By  the  property  of  principal  planes 
C  is  carried  to  Z),  and  by  the  property  of  rays  from  a  point 
in  a  focal  plane  (see  §  35)  DN'  is  parallel  to  BF'. 

N  and  N'  are  two  points,  nodal  points,  which  have  the 
property  that  incident  rays  through  one  of  them  (e.g.  N) 
emerge  parallel  through  the  other  (e.g.  N').  In  this  they 
resemble  the  optical  center  of  thin  lenses.  (Conf.  §  70.) 


THICK    LENSES 


43 


64.  Evidently  HN  =  H' N'  =  F'H'  -  FH. 

[Equal  triangles,  etc. 

Therefore  if  FH  =  F'H',  i.e.  the  focal  distances  the 
same,  due  to  same  media  on  both  sides  of  the  lens,  the  usual 
case  (but  Conf.  §  71),  then  H  and  N  coincide,  likewise  H' 
and  N'. 

Every  incident  ray  through  the  first  nodal  point  emerges 
as  a  parallel  ray  through  the  second  nodal  point.  There- 
fore the  angle  subtended  by  an  object  at  the  first  nodal 
point  equals  the  angle  subtended  by  the  image  at  the 
second  nodal  point,  just  as  in  the  thin  lens  the  angles 
subtended  at  the  center  by  object  and  image  are  the  same 
size. 

In  the  human  eye,  the  second  nodal  point  is  within  the 
crystalline  lens  about  .4  mm.  from  the  back.  (Conf.  §  71.) 

65.  Optical  Center. 


2dsurf. 


C      Center  of       Center    O 
1* surf  ace    of  2*  surf  ace 


Construction.  —  From   the  centers  of  the  two  surfaces 
draw  parallel  rays  and  find  the  point  C  as  shown. 


By  sim.  triangles 


=  -  (left-hand  diag.) 


-  CO       -  r     . 

,        --  (right-hand  diag.) 


CO  = 


s  —  r 


—  7* 


00'  = 


er 


—  r 


(s  -  r  -  e) 


s  —  r 


44 


THICK-LENS    OPTICS 


EXAMPLES 
1.   Neg.  lens,     r  =  2,  s  =  -  3,  e  =  1. 

Ans.     AC  = 


,  A'C  = 
5  5 


2.   Pos.  lens,     r  =  -  3,  s  =  2,  e  =  1. 


Ans.     AC  = 


66.   Construction  for  Nodal  Points. 

Caution.  —  These  constructions  apply  strictly  only  to 
points  near  the  axis,  but  they  serve  to  illustrate  the  prin- 
ciple. (Conf.  remark,  §  61.) 

Draw  a  and  a'  parallel  through  the  centers  0,  0',  giving 
the  points  B,  B'  .  From  the  ray  BB'  construct  the  re- 
fracted rays  c,  c'.  Where  c,  c'  prolonged  cut  the  axis  will 


be  the  nodal  points  N,  Nf.  Where  BB'  cuts  the  axis  is 
the  optical  center,  since  c  is  parallel  to  c',  being  equally 
inclined  to  the  parallel  radii,  a,  a'. 


AC 
A'C 

AO  - 

CO 

er           -  ef 

^^ 

ftr 

=  AA' 
es 

s 
+  AC 
e/' 

-r       /'+/ 
-  Seediag.§65 

f  — 
f 

/x-  1 

-  /tAS 

J  - 
c  — 

ft-  1 
AA' 

ir-i 

;       f'+J 

THICK    LENSES  45 

Hence  the  position  of  the  optical  center  is  fixed  for  two 
given  surfaces  a  distance  e  apart,  since  only  constants 
enter  into  its  value. 

If  the  light  comes  from  the  left,  A  and  A'  interchange 

places,  also  r  and  s,  and  A'C  =  ,       „>  AC  =    ,  ]_ 

67.   Calculation  for  Nodal  Points. 

Evidently  any  ray  passing  (after  refraction)  through  G 
will  enter  and  emerge  in  parallel  lines,  since  the  surfaces 
at  the  points  of  incidence  and  emergence  are  parallel  (being 
perpendicular  to  the  parallel  radii  from  0,  0'). 

Evidently  a  ray  pointing  to  N  before  refraction  will 
after  refraction  emerge  in  a  parallel  direction  as  if  coming 
from  N'. 

N  and  C  are  conjugate  foci  for  the  first  surface,  there- 
fore, §  11, 

[Note.  —  Use  the  diagram  on  the  right  (for  reasons  given 
in  §  21,  note)  until  second  reading.] 


=  —  n 


e/ 

if  light  comes  from  the  left 


46  THICK-LENS    OPTICS 

Therefore    AN  —  dist.  from  first  vertex  to  corresponding 
nodal  point 

-e/ 


(f  -L  fi  _  —  y  ^  li^ht  comes  from  left  J 

Similarly   A'  Nf  =  distance  from  second  vertex  to  corre- 
sponding nodal  point 

e/' 


(  =  — ( f  -L  f '  —   V  ^  light  comes  from  the  left) 

N  N'  =  distance  between  the  nodals 

e/  +  e/'        =  Q-l)  (/  +  /')  +  „( 


E 
E 


if  light  is  from  the  left 


=  e  (r  -  s  +  e)  Q  -  1) 
/A  (r  —  s  +  e)  —•  e 

e  (r  -  s  -  e)  (A*  -  1)  .f  Ught  .g  f  rQm        j  f 
/*  (r  -  s  -  e)  +  e 

e  (neglecting  very  small  terms) 


=  -  e  f  or  glass 

o 


68.  In  computation  check  the  numerical  value  for  N  Nf 
by  the  separate  values,  e  —  AN  —  A'N'j  and  check  by 
graphic  construction,  as  in  §  73. 


THICK   LENSES  47 

Ex.  1.  Show  that  AN:A'N'  has  the  ratio  between 
r  and  s,  the  radii  of  the  surfaces. 

69.   Image  in  One  Nodal  Plane  of  Object  in  the  Other. 

If  P  is  the  object  in  one  nodal  plane  (which  may  be 
outside  the  prism  altogether,  see  §  72,  Ex.  3),  we  can  find 
its  image  in  the  other  nodal  plane  by  tracing  known  rays. 


The  explanatory  details  of  one  diagram  apply  equally  to  the  other. 

The  known  rays  are  rays  through  the  center  of  curvature, 
which  enter  the  corresponding  surface  without  refraction. 

The  ray  PO,  which  is  unrefracted  by  the  first  surface, 
gives  an  image  R  in  the  plane  RC.  The  image  R  becomes 
the  object  for  a  new  image  Q,  made  in  the  nodal  plane  Nf 
by  the  unrefracted  ray  O'R. 

PN       NO   QN'  =  N'O' 
RC  ==  CO'  RC 


Therefore 

PN 
QN' 


Therefore 
PN  = 


PN 
RC 


RC 
QN' 


NO 
CO 


CO' 


CO' 


[Sim.  triangles 


N'O' 


=  1 


[CO       r 
\Cp'  ~  s 


ON 
O'N' 


NO    CO' 
N'O' '  CO 

[Sim.  triangles 

=  -     Sim.  triangles 
s 


48 


THICK-LENS    OPTICS 


Hence  the  object  in  one  nodal  plane  has  an  equal  and 
erect  image  in  the  other  nodal  plane;  i.e.  all  rays  passing 
through  P  in  one  plane  will  pass  through  Q  in  the  other. 
PQ  is  parallel  to  00'.  The  image  in  one  nodal  plane  is 
transferred  horizontally  without  change  of  size  to  the  other 
nodal  plane.  (Conf.  §  60.) 

70.  Lens  separating  Different  Media. 


Diagram  showing  the  paths  of  two  sets  of  rays 
when  the  principal  points  and  the  nodal  points  do 
not  coincide.  Compare  this  with  the  diagram  of 
§  74,  where  the  principal  points  and  the  nodals 
coincide. 

71.  The  human  eye  illustrates  this  case,  the  aqueous 
humor  being  on  one  side  of  the  lens  and  the  vitreous  humor 
on  the  other,  under  which  circum- 
stances the  principal  points  and  the 
nodal  points  are  separated  and  the 
two  foci  are  different.  (Conf.  §  22.) 
The  principal  points  are  very  close 
together  at  H,  about  2  mm.  behind 
the  cornea;  the  nodal  points  almost  as  close  together  at 
N,  about  7  mm.  behind  the  cornea. 

The  anterior  focus  is  at  F,  about  13.7  mm.  in  front  of 
the  cornea,  and  the  posterior  focus  at  Ff,  about  22.8  mm. 
behind  the  cornea. 


THICK    LENSES  49 

Note.  —  Since  the  investigations  of  this  book  are  gen- 
erally for  the  case  where  the 
principal  planes  and  the  nodal 
planes  coincide  (lens  in  air),  the 
term  nodals  has  been  used  indis- 
criminately for  the  coincident  "T 
points  H  and  N. 

72.    Focal  Length  of  Thick  Lens. 

For  parallel  incident  rays,  the  image  by  refraction  from 
the  first  surface  will  (§  12)  be  at  a  distance  /  from  the 
surface,  and 

fJL    —     1  /A 

~^~  =7 

Therefore  the  distance  of  the  first  image  from  the  second 
surface  will  be 

(If  the  light  comes  from  the  left,  this  distance  will  be/  —  e.) 
If  v  =  the  distance  of  second  image  from  second  surface, 
then,  §  11 


f  +  e       v~        s  f 

Therefore  -  -        **      +  ^  -    u.e  +  f 

~  +      " 


Whence  v  =      f  (f  +  e) 

• 


/  =  -    — -  =  focal  rad.  for  1st  surf. 

/'  = — -  =  neg.  focal  rad.  of  2d  surf. 

e  =  thickness  of  lens 

r   =  rad.  of  curv.  of  1st  surf. 

s   =  rad.  of  curv.  of  2d  surf. 

F  =  principal  focus  =  focus  for  parallel  rays 


50  THICK-LENS    OPTICS 

Therefore 

N'F  =  A'F  -  A'N'  =  "" 


=F 


e) 


which  is  called  the  focal  length,  for  reasons  indicated  in 
§57. 

/ 

[  If  the  light  comes  from  the  left,  we  have  F  =      , 

\  /*  (J  ~r  J 

EXAMPLES 
1.   Negative  lens,  r  =  5,  s  =  7,  /*  =  1.5,  e  =  .2. 

Ans.     f  =  ^        =  15,  f  =  -  l-^t  -  -  21,6 


=  .2  +  15  -  21  =  -  5.8,  AN  =  .34,  A'AT'  =  .48, 
F  =  36.2.  Notice  that  both  nodal  points  are  outside  the 
lens.  Light  from  the  right. 

2.  Negative  lens,  r  =  5,  s  =  —  7,  p  =  1.5,  e  =  .2. 

^ns.  Light  from  right.  /  =  15,  /'  =  21,  A  N  =  -.055, 
A'N'  =  .077,  F  =  5.80.  Notice  that  the  nodal  points  are 
both  inside  the  lens  and  close  to  the  surfaces. 

3.  Light  from  right,     r  =  7,  s  =  5,  ft  =  1.5,  e  =  .2. 

Ans.  Positive  lens.  /  =  21,  /'  =  -  15,  A  N  =  -  .451, 
A'N'  =  -  .323,  F  =  -  33.87.  Nodal  points  are  both 
outside  and  behind  the  lens. 

4.  Light  from  right,     r  =  —  7,  s  =  5,  ^  =  1.5,  e  =  .2. 

Ans.  Double  convex  lens.  /  =  —  21,  /'  =  --  15,  AN 
=  -  .078,  A'N'  =  .056,  F  =  -  5.85.  Nodal  points  are 
inside  and  very  near  the  surfaces. 


THICK    LENSES  51 

5.  Light  from  left,     r  =  5,  s  =  7,  i*>  =  f  ,  e  =  .2. 

Ans.    f  =  15,  f  =  -  21,  AN  =  -  .32,  A'  N'  =  -  .45. 
Nodal  points  outside  the  lens. 

6.  Light  from  right,     r  =  —  7,  s  =  —  5,  /*  =  f  ,  6  =  .2. 
Ans.    /  =  -  21,  f  =  15,  A  TV  =  -  .48,  A'  N'  =  -  .34. 

7.  Double  convex  lens,     r  =  —  f  ,  s  =  1,  e  =  J,  /u,  =  f  . 
Light  from  right. 

.    /  =  -  f,    /'  =  -  3,    AN  =  -  A,    A'N'  =  A, 


8.  Double  convex  lens,     r  —  f  ,  s  =  —  1,  e  =  £,  /*  =  f. 
Light  from  left. 

Ans.    /  =  £,  /'  =  3,  A  N  =  A  =  1-58,  A'  N'  =  --  A  = 

-  0.21,  F  =  0.947. 

9.  Negative  lens.     Light  from  left,     r  =  —  |,  s  =  oo, 
e  =  .1,  ft  -  f. 

Ans.    /=-3,    /'  =  oo,    AN  =  0,     A'N'  =  -  A  = 

-  0.0625,  F  =  -  V5    =  "  L875. 

10.  Double  convex  lens,     r  =  —  },  s  =  1,  e  =  },  /x  =  |. 
Therefore  light  from  right. 

Ans.    /  =  -  f  ,     /'  =  -  3,      AN  =  -  &  =  -  0.157, 
A'N'  =  A  =  0.210,  F  =  -  0.947. 

11.  Double     convex    lens,     r  =  —  f  ,     s  =  10,     e  =  J, 
/A  =  f  .     Therefore  light  from  right. 

Ans.    /  =  -  |,  f  =  -  30,   A  N  =  -  Tf  T  =  -  0.0236, 
A'N'  =  AV  =  0.315. 

12.  Double    convex    lens,     r  —  —  f,    s  =  100,    e  =  J, 
ft  =  |.     Therefore  light  from  right. 

Ans.    /  =  -  |,  /'  =  -  300,  A  N  =  -  0.00248,  A'  TV'  = 
.331. 


52  THICK-LENS    OPTICS 

(Examples  10, 11, 12  are  to  show  how  the  flattening  of  the 
lens  causes  the  node  to  approach  one  face.) 

13.  Piano  convex  lens,     r  =  16,  s  =  oo,  e  =  2,  /u,  =  £. 
Therefore  light  from  left. 

Ans.    f  =  48,   /'  =  oo,   AN  =  0,   A' N'  =  |,    F  =  32. 

Notice  that  in  a  piano  convex  the  nodes  are  independent 
of  the  finite  radius.     Ditto,  piano  concave. 

14.  Positive  meniscus,     r  =  10,  s  =  16,  e  =  2,  ^  =  f . 
Therefore  light  from  left  and  lens  convex  towards  the  left. 

Ans.    f  =  30,  /'  =  -  48,   AN  =  -  2,   A' N'  =  -  3.2, 
F  =  48.     Both  nodes  outside. 

15.  Non-curvature  lens,     r  =  10,  s  =  10,  e  =  2,  ^  =  |. 
Therefore  as  in  Ex.  14. 

Ans.    f  =  30,  /'  =  -  30,  AN  =  -  20,  A1 N'  =  -  20, 
F  =  300.     Therefore  as  in  Ex.  14  both  nodes  outside. 

16.  Double    convex    lens,     r  =  10,    s  =  —  16,    e  =  2} 
/u,  =  f .     Therefore  light  from  left. 

Ans.    f  =  30,    /'  =  --  48,     A  N  =  fj,     A' N'  =  -  i§, 
F  =  -2r49°-     Nodes  inside. 

17.  Double  convex  lens,     r  =  —  10,  s  =  16,  e  =  2,  p  = 
f .    Therefore  light  from  left. 

Ans.    f  =   --  30,  /'  =   --  48,    AN  =  i,    A'tf'  =  -  f, 
F  =  —  12.     Nodes  inside. 

18.  Piano    convex,     r  =  GO,    s  =  —  16,   e  =  2,   /x  =  f. 
Therefore  light  from  left. 

Ans.    f  =  oo,  /'  =  -  48,   AN  =  f,   A'N'  =  0,   F  =  32. 
Nodes  inside,  one  tangent. 

19.  Concentric   lens    (Ross   lens),     r  =  3,   5  =  1,   e  =  2, 


THICK    LENSES 


53 


Ans.  f  =  9,  /'  =  -  3,  AN  =  3,  A1  N'  =  1,  F  =  -  4J. 
The  nodes  coincide  at  the  center. 

20.  In  Ex.  8,  if  u  =  2.594,  whence  v  =  1.492,  show  that 
(§  75)  xy  =  0.9472. 

73.  Graphic   Check.  —  To  de- _Z          /     r 

tect   large   errors  check  the   cal-     S*/-; 

culation     by     similar     triangles, 

drawn  to  scale,  in  which  the  sides  \ /' 

are  as  shown. 

A  large   error  will   be   quickly 

detected  in  this  way  before  it  has  time  to  vitiate  the  fol- 
lowing calculations. 

74.  Construction  for  Image  (Conf.  §  25). 


Jl 


C'D  is  parallel  to  AC  (§  63). 
By  sim.  triangles 

u       k       m  F       ,  ,. 

"  -  1  -  T  ~  ^=  -J  (upper  dia 


54  THICK-LENS   OPTICS 

u       k       m  F      .,  ,. 

*  ==  "T :  7  =  F^~v  (1°Wer  di 
Whence 

I  -  -  =  - 

v       u       F 

Hence  the  distances  of  object  and  image  from  the  nodal 
points  obey  the  same  law  as  the'  distances  from  the  lens  in 
the  case  of  thin  lenses,  the  focal  length  being  the  distance 
from  the  nodal  point  of  emergence  to  the  principal  focus. 

The  nodal  planes  take  the  place  of  the  two  coincident 
faces  of  the  thin  lens,  and  the  constructions  and  calcula- 
tions are  carried  on  as  if  the  thin  lens  were  split  and  then 
the  two  edges  of  the  split  separated  the  distance  between 
the  nodal  planes. 

75.   Exercise.  —  From  the  diagram  show  that 


&*  -  F)  (v  -  F)  =  F2 

or,  as  it  is  generally  written,  xy  =  F2,  x  and  y  being  the 
distances  of  the  object  and  image  from  the  focal  points. 

76.  Use  of  Formulae.  —  Decide  upon  the  direction  of 
light  and  give  the  corresponding  signs  to/,/'  (see  §§  36,  37). 
Then  select  the  proper  formulae  corresponding  to  the  direc- 
tion of  the  light.     (Light  from  the  left  makes  the  /  of  the 
positive  lens  +  ,  a  seeming  gain  in  concordance  of  signs.) 

77.  Graphic  Tracing  of  any  Ray  Path. 

This  follows  the  formula  of  §32,  rl— »  $2  ||,  exactly, 
except  that  the  lens  line  is  split  apart  the  distance  between 
the  nodal  planes,  the  points  in  one  nodal  plane  being 


THICK    LENSES 


55 


dragged  horizontally  to  the  other.     The  order  of  the  letters 
indicates  the  order  of  construction,  x  being  the  line  sought. 


78.  Since  the  nodal  planes  are  really  plane  surfaces, 
their  intersections  with  the  paper  will  be  straight  lines,  as 
drawn  in  the  diagrams.  Therefore,  having  the  nodals  and 
foci  of  two  lenses  given  in  position,  we  can  find  the  nodals 
of  the  combination  by  §  61,  by  taking  the  initial  rays 
horizontal. 


Jf     JV  F 

Diagram  showing  the  tracing  of  a  ray  through  two  successive  lenses, 
the  principal  planes  of  each  lens  being  indicated  by  letters  in  horizon- 
tal lines. 

Example  1.  —  Try  this  on  the  combinations  of  §107, 
Exs.  4,  8. 

Example  2.  —  See  §  95,  Ex.  2. 

These  two  sections  are  an  extension  of  the  principles  of 
§  32,  and  are  equally  important  in  the  application  to  nodal 
planes. 

ANALYTICAL  INVESTIGATION  1 

79.    The  previous  investigation  has  assumed  some  facts 

JThe  remaining  sections  of  this  chapter  are  for  those  inquisitive 
readers  who  desire  a  somewhat  more  rigorous  logic  and  less  depend- 


56 


THICK-LENS    OPTICS 


as  self-evident.  The  investigation  of  this  section  is  for 
the  purpose  of  putting  these  facts  on  a  more  strictly  logical 
basis,  to  meet  the  criticism  to  which  the  preceding  sections 
might  be  open  to  the  casuist. 

80.   Before  entering  upon  the  discussion,  we  give  some 
preliminary  principles. 

y  =  dist.  above  (or  below)  the  x  axis,  of  an  arbitrary 
point  on  the  line 

x  =  distance  of  the  point,  to  right  or  left  of  the  y  axis 
a,  b  =  corresponding  distances  for  some  fixed  point 
x  and  y  have  many  values,  one  for  each  point 
a  and  b  are  constant,  fixing  some  definite  point 


By  sim.  triangles 


x  —  a 


d 


m 


or,  as  it  is  generally  written 


y  —  b  =  m  (x  —  a) 
This  is  called  the  equation 
of  the  line  referred  to  the 
axis,  since  x  and  y  taken  in 
corresponding  values  fix  any 
point  on  the  line.  Their 
values  could  be  used  to 
plot  points  on  the  line;  or 
corresponding  values 
measured  from  a  point  on  the  line  will  satisfy  the  equation. 
From  the  equation  we  can  locate  the  line  by  assuming  a 
value  for  x  and  calculating  the  corresponding  value  of  y, 
and  then  plotting  the  two  values,  thus  locating  a  point, 
and  so  on.  In  other  words,  an  equation  of  a  line  gives  us 
a  clue  as  to  where  the  line  lies. 


8 

/^        d 

y 

!> 

' 

\ 

>X^ 

b 

i 

s/6 

a 

i 

x  axis 

ence  upon  intuition.     They  can  safely  be  omitted  by  those  not  inter- 
ested, without  destroying  the  continuity  of  the  text. 


THICK    LENSES 


57 


The  advantage  of  the  equation  is  that  we  can  operate 
upon  the  equation  algebraically  and  then  interpret  the 
result  geometrically,  without  going  through  all  the  pecul- 
iarities of  a  geometrical  diagram. 

81.    General  Equation  of  a  Refracted  Ray. 


©,  (2),  and  (D  represent  the  ray  before,  during,  and  after  refraction 

For  rays  through  points  near  the  vertex  A,  so  that  the 
point  of  incidence  is  practically  over  A, 

1.  The  equation  to  line  ©  is  (§  80): 

y  —  b  =  m  (x  —  OA) 

2.  For  line  (D 

y  -  b  =  m'  (x  -  OA)  or  (y  -  6')  =  m'  (x  -  OA') 

3.  For  line  (D      y  -  b'  =  m"  (x  -  OA') 

4.  From  equation  2,     b  -  b'  =  m'  (OA'  -  OA}  =  m'e 
By  §  4  sin  r®  =  ^  sin  r(D 

[r©  means  the  angle  between  r  and  ©,  etc. 

c  _  sin  r© 
r       sin  c© 


But  by  §  6 


Therefore  -  sin  c©  =  sin  r©  =  /*  sin  r© 


5. 


d   . 

=  P>  -  sm 
r 


[d      si 
r  ~s 


sin  r(D 


58 


THICK-LENS    OPTICS 


Therefore      c  =  b  +  m  (OC  -  OA)       [Eq.  1  taking  y  =  c 

=  b  +  mr 

Similarly      d  =  b  +  m'r 
Therefore     (b  +  mr)  sin  c®  =  p  (b  +  m'r)  sin  d(j) 


or  6  +  mr  =  p  (6  +  ra'r) 


Since  sin  c(D  =  sin 


prac- 


tically, the  two  angles  being 
nearly  90°  each 


6.   Whence     pmf  =  m  — 6  =  m  —  bu 


Making  —    -  =  u 


7.   Similarly  pim'  =  m"  —  6V 

E  =  — ,  pi  =  index  of  refraction  for  2d  surface 
» 
s  =  radius  of  2d  surface 

XT       */       -,    .   m  —  bu          ,  f          eu\    .   me        rri 

Now  b'  =  6  H e  =  6  (  1 )  H [Eqs.  4,  6 


8.     =  gb  +  hm 


r. 


Lr>    X.L-         i          eu  e         7 

Putting  1 =  gr ,  -  =  h 


,t  ,    ,    ,    ,  (1  —  eu)    .    men' 

m"     =  /*im'  +  bu'  — 


m  —  bu 


[Eqs.  8,  7 


9. 


- 


p. 


meu 


10.     =  kb  +  Im 


^  ...      7         ,        pi 
Putting  k  =  u  -- 


euu 


=  ug  —  u  - 


Z  = 


eu 


THICK    LENSES 


59 


If  6  =   Y-m(X-OA) 


X,  Y  being  co-ordinates  of  the 
point  on  the  line  which  is 
considered  as  the  source  of 
the  ray 


then 


6'  =  g  Y  +  m  (h  -  g  •  X  -  OA) 


m 


From  these 


Whence 


k  Y  +  m  (I  -  k  •  X  -  OA) 

m"  -  k  Y 
m       l-k(X-OA) 


h-g(X-OA) 


m"  (x  -  OA' 


11.    Or 


h-g(X-OA)\ 
+  l-k(X-OA))          [Eq'3 

*  mn(x      OA,  ,    h-g(X-OA)\ 

y~l*[l-k(X-OA)}~        \X  ^l-k(X-OA)) 

Since  gl  —  hk  =  ~ 


the  equation  of  the  emerged  ray  in  terms  of   X,   Y,  the 
co-ordinates  of  the  source. 

82.   If  X  be  taken  such  that      Z  -  k  (X  -  OA)  =  ^ 


that  is 
then  when 
x  =  OA'  - 


X  =  OA  + 


=  OH,  say 


h-g(X  -OA) 


60 


THICK-LENS    OPTICS 
l-^ 


=  OA'--    \h- 

Mi 


I— t 


X- 


[Si 


ince  gL  —  hK  =  — 


=  OA' 
we  will  have 


say 
=    F 


and  in  the  planes  of  these  two  points  (H  and  H')  the  object 
(F)  and  the  image  (y)  are  equally  distant  from  the  axis 
of  the  lens;  the  rays  are  transferred  horizontally.  (Conf. 
§§  62,  69.) 

These  points  are  called  principal  points,  and  perpendic- 
ular planes  through  them  principal  planes. 

83.    If  X  be  taken  such  that  I  -  k  (X  -  OA)  =  1, 


whence 
then  when 


X  =  OA  + 


l-l 


ON,  say 


x  =  OAf  -  \h-g(X-  OA)\  =  OA'  -  (h  -  gl-jl 

I-  1 


X  -  OA  = 


THICK    LENSES  61 


and  when   Y  =  0,  then  also  y  =  0,  and 

m"  -  k  Y 
~  I-  k(X  -OA) 

since  kY  =  0  and  I  -  k  (X  -  OA)  =  1 

84.  Since  m  =  m",  the  rays  before  refraction  and  after 
refraction  are  parallel  (Conf.  §  63),  and  the  image  is  not 
deflected  so  long  as  this  point  N'  is  not  moved.     (Conf. 
§59.) 

The  points  N  and  N'  are  called  the  nodal  points. 

85.  If  /MI  =  /A  (e.g.  air  on  both  sides),  then  H  and  Hr 
coincide  respectively  with  N  and  N'.     (Conf.  §§  64,  71.) 

86.  If  m"  =  0,  i.e.  the  ray  is  parallel  to  the  axis  after 
refraction,  then  from  eq.  10 

6  =  —  7-  m 
k 

and  the  equation  of  the  incident  ray  is 

y  +  -j-  =  m  (x  —  OA)    or    y  =  m  (  x  —  OA  --  ^  J 

87.  If  we  also  take  y  =  0,  so  as  to  find  where  the  ray 
crosses  the  axis,  then 

A*i       erf 

x  =  OA  +  \  =  OA  +  -    -£  --  ^  -  7        [Eq.  10 
k  ,          PI       euu 

u   —  u  --- 

f-          M- 
=  OF,  say 

88.  If  m  =  0,  i.e.  the  incident  ray  parallel  to  the  axis, 
then  from  eqs.  8,  10 


62  THICK-LENS    OPTICS 


and  the  equation  of  the  refracted  ray  becomes,  eq.  3 
y  -  Q^f  =  m"  (x  -  OA')  or  y  =  m"  (x  -  OA'  +  | 

whence  if  also  y  =  0,  in  order  to  find  where  the  ray  crosses 
the  axis,  then  the  distance  to  the  crossing  point  is 

/*i  _  eu 

x  =  OA'  -  \  =  OA'  ---  ^  --  ^  --  T  =  OF',  say. 
k  ,  /u-i       euu'  '     J 

u'  —  u  —  -- 
/*          ^ 

F  and  F'  are  called  the  focal  points. 
89.    For    /AI  =  /A,  the  usual  case 


OF  =  OA+ 


. 
I*,  (u   —  u)  —  euu 

OA-       fU'~<» 

/^  (/  +  /'-«) 

OF'=OA'  ~ 


,        . 
(u   —  u)  —  euu 

OA'+       '- 


^ 

OF'  -  ON'  =  OA'  -  £  -  (oAf  +  ^-r-^1  =  ON  -  OF 
-  1 


k 

-fi  ff 


-u    -euu 


=  /^ 


equivalent  focal  length  of  the  lens 

(Conf.  §  72) 


THICK  LENSES 


63 


90.   From  eq.  11 


-  g(X  -  OA)\ 


l-k(X-OA) 


=  m"  (x  -  t)  + 


l-k(X-OA) 
Y 


I  -  k(X  -  OA) 

where  £,  rj  are  evidently  (since  x  =  £,  y  =  rj  satisfies  the 
equation)  on  the  ray  d),  and  evidently  dependent  only 
on  X  and  Y  and  not  on  m,  6;  that  is,  every  ray  through 
X,  Y  (the  object)  passes  through  £,  >/  (the  image). 

From    OA  =  ON  -  -~,  OA'  =  ON'  -  ^-^ 


substituting  these  values  in  the  expressions  for  £,  17,  we  get 
ON  -X  Y 


whence 


or 


1  +k(ON  -  X) 
1  1 


X-ON 


k- 

—    —   '•'    —   ~Fi 

P       Pi  F 


"  1  +k(ON  -  X) 
=  -k 


where  p  =  distance  from  node  N'  to  the  image  £,  and 
pi  =  distance  from  the  object  X  to  the  node  N.  This 
shows  that  the  nodal  distances  to  object  and  image  obey  the 
same  laws  as  the  thin  lens  distances.  (Conf.  §  74.) 


CHAPTER   IV 

COMBINATIONS    OF    LENSES 

91.  Thin  Lenses  in  Contact. 

For  the  first  lens  =  T 

Vi          U         fi 

for  the  second  lens  =  7 

v*       Vi      /2 

whence,  by  addition  =  7-  +  7 

v2      u      /i      fz 

1 

~  F 

/i,  /2  =  focal  lengths  of  the  lenses 
u  =  dist.  of  object  from  1st  lens 
Vi  =  dist.  of  1st  image  from  1st  lens  and  of  2d 

object  from  2d  lens 
v2  =  dist.  to  image  formed  by  2d  lens 
F   =  focal  length  of  combination 

Example.  —  A    +    lens,    2    in.    focus,    is    cemented    to 
a  —  lens,  9  in.  focus.     What  is  the  equivalent  focus? 

Ans.     Equivalent  focus  =  2f . 

92.  For  a  third  lens,  similarly 

I        I        I        1  _L_  I        1 

V,  W"/1+  /2  /3~F 

The  power  of  the  combination  is  the  sum  of  the  powers 
of  the  components.     (Conf.  §  42.) 

For  powers  of  lenses  not  in  contact,  see  §  96. 

64 


COMBINATIONS    OF    LENSES 


65 


93.   Thin  Lenses  not  in  Contact. 

Taking  A,  a  point  on  the  refracted  ray  and  in  the  front 
focal  plane  of  the  second  lens,1  we  can  find  its  image,  C, 
through  the  second  lens  by  known  rays,  as  shown  (or  by 
§  32) .  But  any  other  ray  through  A ,  as  A  B,  must  go  through 
the  same  image  point,  and  thus  we  get  the  direction  BC  for 
this  A  B  ray  after  it  has  been  refracted  by  the  second 


1*1 

Horizontal 

ens 

Z* 

A 

ens 
£ackfocu$  of 
Z'lens   Back  focus  of 
^x^              combtnatLOn 

ray 

^\^:  —  —  __ 

ofZ^lens          focus  tf^~^ 

IUn°        ^ 

^^-Im'age  ofJlby  . 
^"^  known  rays  ••  rwri  zon- 
tal  and  thru  center 

Diagram  illustrating  general  case  of  two  thin  lenses,  /*  not  equal 
to  m;  hence  front  and  back  focal  length  of  second  lens  will  have 
different  values.  Introduced  for  the  purpose  of  getting  a  general 
rule  of  construction  for  use  in  the  next  diagram,  indicated  by  the 
order  of  the  letters. 

lens.     The  point  D,  where  it  meets  the  axis,  is  the  back 
focus  of  the  combination  for  incident  parallel  rays. 


EXAMPLES 

1.  A  candle  is  held   1  foot  in  front  of  a  convex  lens, 
giving  an  image  on  a  screen  4  inches  behind  it.     A  con- 
cave lens  is  now  placed  in  contact  with  it,  and  the  screen 
must  be  moved  8  inches  further  away  to  get  the  image. 
What  is  the  focal  length  of  the  negative  lens?     Check  by 
§  32.     Ans.     -  6. 

2.  A    convex   lens   of   16  cm.   focus,   in    contact    with 
a  negative  lens,  gave  a  focal  length  of  48  cm.  for  the 

1  In  order  to  have  its  distance  from  the  lens  a  definite  and  significant 
value. 


66  THICK-LENS    OPTICS 

combination.     What  is  the  focal  length  of  the  negative 
lens?     Ans.     -  24. 

3.  A  concave  lens  of  —  8  cm.  is  combined  with  a  con- 
vex lens  of  +  6.     What  is  the  focus  of  the  combination? 
Ans,  24. 

4.  By  the  method  of  §§  32,  33,  show  that  a  compound 
microscope   (two  positive  lenses  more  than  the  sum  of 
their  focal  distances  apart)  must  have  the  object  without 
the  focal  distance  of  the  objective.     (Conf.  §  33,  Ex.  2.) 

5.  When  the  distance  outside  the  focus  in  Ex.  4  becomes 
infinity,  the  distance  apart  of  the  lenses  becomes  the  sum 
of  the  focal  radii,  and  we  have  the  celestial  telescope. 

6.  Show  by  §§  32,  33  that  a  positive  lens  followed  by 
a  negative  lens,  the  distance  apart  being  the  difference  of 
the  foci  or  less,  will  give  a  virtual  image.     (Ordinary  opera 
glass.) 

7.  If  the  distance  apart  in  Ex.  6  is  greater  than  the  differ- 
ence, there  results  a  real  image.     (Telephoto,  §§  113,  29.) 

8.  In  the  Huygens  eyepiece  (field  lens  focus  =  3/,  eye 
lens  focus  = /,   distance  between  lenses  =  2/),  show  by 
§  32  that  rays  incident  upon  the  front  lens  pointed  toward 
a  point  between  the  lenses  f  /  from   the   front  lens  will 
emerge  from  the  second  lens  parallel  to  the  axis;  that  is, 
will  give  a  virtual  image. 

9.  In   the   Ramsden   eyepiece  (field  lens  focus  =  eye 
lens  focus  =  /,  distance  apart  of  lenses  =  f  /),  show  by  §  32 
that  parallel  rays  incident  upon  the  front  lens  converge 
to  a  focus  //4  beyond  the  back  lens. 

10.  By  §§  32,  25,  etc.,  show  that  the  second  lens  of 
Ex.  6  reinverts  the  image  made  by  the  +  lens,  thus  giving 
a  final  erect  (virtual)  image. 

11.  Similarly  show  that  in  Ex.  7  the  real  image  is  kept 
inverted. 


COMBINATIONS    OF    LENSES 

94.   Back  Focal  Distance  for  Two  Thin  Lenses. 

For  construction  of  diagram,  see  §  93. 

-  e  d       _d 

e  ~  d 


67 


By  similar  triangles 


/i  +  /2  -  e 


Back  focus  of 
combination 


g  ~  f2 


Diagram  showing  the  construction  for  back  focal 
distance  when  Hi  =  /*.  The  letters  A,  B,  C  designate 
the  same  points  as  in  the  preceding  diagram. 


dis- 


Therefore  33  =  back  focal  distance  of  combination 
tance  from  lens  to  focal  point 

/«  (/i  ~  e) 

/1+/2-6 

95.   Equivalent  Focus  for  Two  Thin  Lenses.  —  The  two 


lalens 


^Mdal  point  of  the 
combination 

Backjbcus  of  combination 

Backfocusof2Alens 
Focus  ofi*len& 


- 


thin  lenses  act  like  the  two  surfaces  of  a  thick  lens,  and, 
like  them,  have  their  corresponding  nodal  points  deter- 
mined in  the  same  way  (see  §  61),  remembering  that  the 


68  THICK-LENS    OPTICS 

surfaces  are  now  vertical  plane  surfaces.     Add  the  con- 
struction of  §  61  to  the  diagram  of  §  94. 

For  construction  of  diagram,  Conf.  §§  93,  94,  61. 

By  similar  triangles  (lightly  shaded  in  the  diagram  for 
the  second  set) 

€  d       N' 


/!+/»-€       e    '  f, 


Therefore       N'  =  -f — ~. =  nodal  distance  from  corre- 


Ji  +  J2       e  spending  lens 


Therefore        F  =  33  +  N'  =  F-vV-^-  + 


/!/« 


-6     '    /!+/,- 


=  equivalent  focal  length 
Ji  -r/2  of  ^g  combination 

=  distance  from  nodal  point  to  focal  point. 
If  we  take  into  account  the  direction,  we  get 


.      . 

and  similarly 

N  -          fl€ 

1M      -  -    j.        .       f 

/i  +  /i  —  € 

This  section,  with  §  61,  enables  us  to  find  the  nodal 
points  of  a  combination  of  lenses. 

Ex.  1.  In  the  Huygenian  eyepiece  (field  lens  focus 
=  3/,  eye  lens  focus  =  2/,  c  =  2/),  show  that  the  front 
focal  distance  (found  by  parallel  rays  through  the  eye 
lens,  from  right)  is  -  }/.  (Conf.  §  93,  Ex.  8.) 

2.  In  the*  Ramsden  eyepiece  (/i  = /2,  e  =  f/i),  show 
that  the  back  focal  length  is  J  f\.  That  is,  the  combination 
has  the  properties  of  an  ordinary  convex  lens  of  //4  focus. 
(It  has  the  advantage,  however,  of  being  approximately 
achromatic.)  (Conf.  §  93,  Ex.  9.) 


COMBINATIONS   OF    LENSES  69 

3.  By  the  method  of  §  61,  show  that  if  two  thin  positive 
lenses  lie  with  crossed  foci,  and  each  lens  within  the  focus 
of  the  other,  the  resulting  foci  of  the  combination  will  lie 
outside  the  lenses,   and  both  nodal  points   between  the 
lenses. 

4.  In  the  preceding  example,  if  each  lens  is  without  the 
focus  of  the  other,  then  both  foci  of  the  combination  are 
between  the  lenses,  and  both  nodals  are  outside,  and  the 
nodals  are  crossed. 

5.  By  §§  32,  25,  etc.,  show  that  a  microscope  composed 
of  a  J  in.  objective  and  a  1  in.  eyepiece,  6  inches  apart, 
for  a  person  of  8  in.  vision  must  have  the  object  |J  in 
front  of  the  objective. 

EXAMPLES 

1.  Two  positive  lenses  with  a  common  focal  length  of 
0.05  m.  are  0.05  m.  apart.     What  image  results  of  a  disc 
0.01  m.  in  diameter  placed  0.1  m.  distant? 

Ans.  A  real  image  0.025  m.  beyond  the  second  lens. 
Diameter  of  image  =  0.005  m. 

Note  the  crossing  of  the  nodes.  Draw  a  diagram  and 
compute  the  magnification  by  similar  triangles.  Conf. 
diagram  of  §  74,  mutatis  mutandis. 

2.  (Microscope   ocular)    Field   lens,  f\  =  2J,    eye   lens, 
/2  =  If,  c  =  2i.     Show  graphically  (see  §§  95,  33,  77)  that 
the  nodes  cross,  N'  to  just  behind  (right  of)  the  field  lens, 
N  to  about  an  inch  behind  (right  of)  the  eye  lens  (light 
from  left);   that  the  posterior  focus  almost  coincides  with 
the  focus  of  the  field  lens ;  the  anterior  focus  not  so  closely 
with  the  anterior  focus  of  the  eye  lens.     Calculate  these 
results  by  the  formulae  of  §  95. 

96.   Powers  of  Thin  Lenses  not  in  Contact. 
From  §  95 


70 


THICK-LENS   OPTICS 


Pi   +  P2   - 


=  power  of  the  combination 

[Pi,  P2  =  powers  of  the  thin  lenses 
c  =  distance  apart 

Example  1.  —  pi  =  3  D,  p2  =  5  D,  c  =  .025  m. 

.'.  Equiv.  power  =  3  +  5  -  15  X  0.025  = 

7.625  diopters 
Example  2.  —  pi  =  p2  =  +  12  D,  e  =  .02  m. 

.'.  Equiv.  power  =  12  +  12  -  (12  X  12  X  .02)  = 

21.12  diopters 
97.  Back  Focal  Distance  for  Light  from  the  Right. 


Diagram  constructed  exactly  as  in  §  94,  mutatis  mutandis. 

f i  +  e  ^  d          & 

By  sim.  triangles        ^       , 

•  Jl     I     /2T" 

Whence  i 


e       d  +  g       /2 
/2  (/i  +  e) 


98.   Equivalent  Focus  for  Light  from  the  Right. 

/1/2 


Similarly  to  §  95 


rr    __ 


COMBINATIONS    OF    LENSES 


71 


Note  the  change  of  sign  of  e  in  the  formulae  when  the 
light  comes  from  the  right,  due  to  the  fact  that  e  is  nor- 
mally positive. 

99.  Back  Focal  Distance  for  Two  Thick  Lenses. 


2dlens 


Tocus  of  combination 

Focus  of2dlens 


Focus  of 
J*len* 


Diagram  showing  construction  for  back  focus  of  thick  lens,  on 
the  same  principle  as  the  preceding  diagrams;  but  note  the  horizontal 
transference  between  the  nodal  planes  in  the  second  lens.  The  nodal 
planes  are  designated  by  N,Nf.  For  construction  see  §  94. 


By  similar  triangles 


d      d' 


whence  33  =  back  focal  distance 


Ji 


(with  light  from  the  left) 


=  distance  from  posterior  nodal  point  of  lens  to 

focal  point 
100.   Equivalent  Focal  Length  for  Two  Thick  Lenses. 

Similarly,  as  in  §  95 

/1/2 


F  = 


(with  light  from  the  left) 


.+/•-.« 

=  distance  from  posterior  nodal  point  of  the 

system  to  the  focal  point 

remembering  that  F  is  measured  from  the  nodal  plane  of 
emergence  for  the    combination,   just   as  /2   and  /i   were 


72  THICK-LENS    OPTICS 

measured  from  the  nodal  planes  of  emergence  of  the  cor- 
responding lenses. 

Hence     F  =  ,    /!/2  .    -  (with  light  from  the  right) 

Jl   T~  J2   ~T    e 

101.   Nodal  Distances  for  Thick  Lenses. 

As  in  §  95,  N'W  =  distance  between  Nf  of  second  lens 
and  nodal  point  of  emission  of 
the  combination,  taking  account 
of  direction 


=  distance  from  N  of  first  lens  to 
nodal  point  of  incidence  of  the 
combination 


/!+/.-€ 

102.  Graphic  Check.  —  Check  for  large  errors,  by  sim- 
ilar triangles  as  in  §§  73,  68,  except  that  there  is  no  /*  in 
the  formula. 

103.  Equivalent  Thickness  of  Thick-Lens  Combination. 
yiW  —  distance  between   the   nodal  points  of  the  com- 
bination 

=  e  +  NN'  (1st  lens)  +  N  N'  (2d  lens)  -  -$  -  $ 

a         a 

[d=fi+f*-  € 

=  NN'  (1st  lens)  +  N N'.  (2dlens)  -  ~ 

a 

=  NN'  (1st  lens)  +  N  N'  (2d  lens)  -  -^ 

/1/2 


COMBINATIONS    OF    LENSES  73 

F       =  focal  length  of  Combination 
/i,/2  =  focal  lengths  of  components 

=  distance  between  the  Nr  of  the  one 
component  and  the  N  of  the  2d 
component 

This  value  should  be  used  as  a  check  in  the  computation 
to  compare  with  the  value  derived  from  the  diagram  by 
introducing  the  various  values,  A  N,  etc.  It  must  be  used, 
of  course,  in  connection  with  the  antecedent  check  of  §  68. 

Example  l.—fi  =  4,  /2  =  3,  N  N'  (1st  lens)  =  .15,  2dlens 
=  .2,  c  =  1.5. 

Ans.     F  =  2.18,  WW  =  -  0.059. 

104.  Power  of  Thick-Lens  Combination. 

From  §100       P=-^  =  T~^T'~rT  =  P^^~P^~  PW* 
f       Ji       J2      /i/a 

If  both  lenses  are  +,  increase  of  c  increases  the  equiva- 
lent focal  length  and  reduces  the  equivalent  power. 

If  one  lens  is  — ,  so  that  the  term  —  p\p&  becomes 
positive,  increase  of  e  will  shorten  the  equivalent  focal 
length  and  increase  the  power. 

The  value  of  WW  (§101)  shows  that  the  equivalent 
thickness  of  a  combination  of  two  +  lenses  is  reduced  by 
separating  them,  and  may  become  zero  or  negative  if  e  is 
large  enough;  i.e.  the  two  nodal  planes  will  cross  each 
other,  as  is  the  case  in  many  camera  lenses  (see  Ex.  7, 
§  106),  microscope  oculars,  etc. 

RESUME 

105.  Notice  that  the  thick  lens  acts  like  a  thin  lens  with 
its  surfaces  (plane  and  perpendicular  to  the  axis,  because 
we  are  considering  only  points  near  the  axis)  split  apart 
and  separated  the  distance  between  the  nodal  (see  §  64) 


74  THICK-LENS    OPTICS 

points,  so  that  if  we  should  make  the  construction  for  a 
thin  lens  and  then  split  the  diagram  along  the  lens  line 
and  pull  it  apart  the  distance  between  the  nodes,  we  would 
have  the  appropriate  diagram  for  the  thick  lens.  We  call 
this  the  equivalent  thin-split.  The  nodal  distance  can,  of 
course,  only  be  found  by  means  of  the  equations  of  §  67, 
after  e  has  been  decided  upon. 

A  combination  of  two  lenses  again  acts  like  two  thin 
lenses  (with  separated  faces,  the  nodal  distance)  removed 
from  each  other  the  distance  (c,  see  §  95)  between  the 
posterior  nodal  point  of  the  first  lens  and  the  anterior 
nodal  point  of  the  second  lens  (see  diagram  of  §  99),  this 
being  the  distance  between  the  inner  faces  of  the  thin 
lenses  before  the  faces  were  split  apart  to  act  as  a  thin- 
split.  This  distance,  e,  can  be  assumed  as  we  please,  and 
then  from  the  equations  of  §  101  we  can  calculate  the  nodal 
points  which  act  as  the  front  and  rear  face  of  the  new 
equivalent  thin-split  lens.  And  so  on. 

That  is,  we  determine  the  equivalent  thin-split  lens  of 
the  individual  lenses,  and  then  the  equivalent  thin-split 
lens  of  the  new  combination,  and  so  on,  the  final  thin-split 
lens  being  the  equivalent  of  the  combination. 

Notice  that  in  the  equations  of  §§  94-100,  /*  has  dis- 
appeared because  the  medium  between  the  two  thin-split 
lenses  is  air  (for  which  /*  =  1)  or  its  equivalent,  even  if  the 
lenses  be  in  contact. 

106.   Use  of  Formulae  for  Combined  Lenses. 


(a)  Find  /,  /',  F,  A  N,  A'  N'  for  each  lens.     (Conf.  §  76.) 


f,  §  15 

,  §§  16,  72 
A  N,  A'  N',  §67 


COMBINATIONS    OF    LENSES  75 

/,  /'  =  focal  radii  for  surface  refraction. 

e  =  thickness  of  component  lens. 

F  =  focal  length  of  the  lens  (or  later  of  the  combination, 
or  thin-split  lens).  (§§  16,  72,  100.) 

A,  A'  =  anterior  and  posterior  vertex  of  lens  (or  com- 
bination of  lenses). 

'AN,  A' N'  =  nodal  distances  from  anterior  and  pos- 
terior vertex  of  lens  (or  later  of  the  combination).  (§§  67, 
101.) 

(ft)  Select  /i,  /2  for  the  combination,  then  find  (decide 
upon)  e,  and  then  compute  F,  W,  W  for  the  combination. 
(§§  100,  101.) 

9fl}  <$lf  =  nodal  distances  from  anterior  (of  the  first 
lens)  and  posterior  (of  the  second  lens)  nodal  points  of 
the  components  to  the  corresponding  nodal  points  of  the 
combination. 

/i,  /2  =  (used  in  the  formula  for  the  combination)  the 
F's  of  the  components. 

e  =  distance  between  the  N'  of  the  anterior  component 
and  the  N  of  the  posterior  component.  €  must  be  taken 
negative  when  the  nodals  cross.  To  calculate  €  locate 
the  nodals  roughly  on  a  diagram,  with  the  distances  noted 
thereon,  and  then  derive  the  value  of  e. 

(y)  Then  find  A  N  -+-  9?  =  distance  of  anterior  nodal 
point  of  combination  from  the  anterior  vertex,  A'  N'  +  9?' 
=  distance  of  posterior  nodal  point  of  combination  from 
the  posterior  vertex  of  the  combination.  A  N  +  %l  of 
component  =  A  N  of  the  (next)  combination,  A'  N'  +  W 
of  the  component  =  A'  N'  of  the  combination.  (See  Exs, 
7,8.) 

Do  not  fail  to  use  the  checks  of  §§  14,  68,  73,  102,  101,  107. 

Repeat  this  for  the  combination  of  this  combination  with 
some  other,  and  so  on. 


76  THICK-LENS    OPTICS 

Great  care  must  be  taken  in  regard  to  +  and  —  signs. 
Here  most  of  the  errors  occur. 

107.  Graphic  Construction  (only  available  when  there  is 
not  too  great  disparity  between  the  numerical  values  used) 
after  the  foci  and  nodals  of  the  component  lenses  have 
been  located  (based  upon  the  calculations  of  §  16  for  thin 
lenses,  and  §§  67,  72  for  thick  lenses,  because  owing  to 
the  disparity  of  the  values  used  the  intersections  in  the 
graphic  work  are  too  acute  to  be  of  service). 

Graphic  construction  is  extremely  valuable  to  check  up 
which  side  of  the  nodals  (which  limit  e)  the  new  nodals  of 
the  combination  lie.  Does  not  need  to  be  to  scale  so  long 
as  the  relation  of  greater  and  less  is  preserved. 

IN  THE  CASE  OF  THIN-LENS  COMPONENTS 

Find  the  equivalent  focus  of  the  combination  by  starting 
with  a  horizontal  ray,  its  first  refraction  being  found  by 
§§  25-30.  Where  the  refracted  ray  strikes  the  second  lens, 
find  its  new  course  by  §  32.  The  intersection  of  this  new 
course  with  the  axis  will  be  the  focus  of  the  combination. 

Find  the  nodals  (principal  points)  by  the  principle  of 
§61  (illustrated  in  the  diagram  of  §  95). 

IN  THE  CASE  OF  THICK-LENS  COMPONENTS 

Find  the  equivalent  focus  of  the  combination  by  starting 
with  a  horizontal  ray,  its  first  refraction  being  found  by 
§  74.  Where  the  refracted  ray  strikes  the  nodal  line  (inci- 
dent, look  out  for  crossed  nodals)  of  the  second  lens,  find  its 
new  course  by  §  77.  The  intersection  of  this  new  course 
with  the  axis  will  be  the  focus  of  the  combination. 

Find  the  nodals  of  the  combination  by  the  principle  of 
§  61,  prolonging  the  last  ray  (second  refraction)  back  to 
its  intersection  with  the  original  horizontal  ray. 


COMBINATIONS    OF    LENSES  77 

EXAMPLES 

1.   Two  thin  positive  lenses,  2  inches  apart,  of  focal 
lengths  6  and  9. 

2'6         12 


A'N'  +  W  =  0  + 
F  = 


6  +  9-2       13 

-2-9  18 

6  +  9  -  2  ~        13 
6-9  56 


6  +  9-2       13 
The  nodals  are  between  the  lenses. 

2.   Thin  lenses,   5  inches  apart,   a  positive  of  6  focal 
length  and  a  negative  of  —  2. 

A  *r    .    c»  5  •  6  +  30 

AN  +  K     -0  +  6_2_5=      -y 

A'#'  +  ft'  =  o  +  75i~2^  =  -  10 


F- 


Notice  that  the  nodals  are  outside  the  combination  and 
far  in  front,  and  that  the  back  focal  distance  is  only  2. 
Distance  between  nodals  =  25. 

3.  Same  as  in  Ex.  2,  but  the  combination  reversed. 

A  N  +  ft  =  10  A'N1  +  ft'  =  30 
Back  focal  distance  =  42  F  =  12 
Distance  between  nodals  =  25 

4.  Two  thick  lenses. 

First  lens,     r  =  6,  s  =  —  4,  e  =  3,  /*  =  1.5 
/  =  18,  /'  =  12,   A  N  =  J,   A'N'  =  -  |  =  -  .889 

F  =  -V5  =  5.33,  NN'  =  I 
Second  lens,     r  =  —  4,  s  =  —  2,  e  =  1,  ft  =  f 


78  THICK-LENS   OPTICS 

/=  -  ¥,  /'  =  ¥,  AN  =  n  =  1-05,     A'N'  =  «  =  .525 
F  =  VT°-  =  5.614,  NN'  =  T9s 

Combination.     Assume  e  =  2,  thus  making  the  lenses  0.062 
apart. 
/i  =  5.33,  /2  =  5.614 

_5.33_XJ5.614_       29.939 
5.33  +  5.614  -  2    "  8.947 

A  N  +  ft  =  1.33  +  233  =  1.33  +  1.19  =  2.52 
(located  in  the  first  lens) 

A'N'  +  W  =  .525  +  -  "    847'        =  «525  ~  L25  =  ~  - 
(in  the  second  lens) 
9W  =  0.804 

5.  Two  thin  lenses,  4  inches  apart,  a  positive  of  6  focal 
length,  and  a  negative  of  —  6  focal  length,  the  positive 
lens  in  front. 

AN  =  -  6,  A'N'  =  -  6,  F  =  9 

The  nodals  are  outside  the  lenses,  6  and  2  respectively, 
from  the  front  face  of  the  positive  lens.     yiW  =  4. 

6.  Lens  of  Ex.  8,  §  72,  0.25  in  front  of  lens  of  Ex.  9. 
Whence 

/i  =  0.947,  /2  =  -  1.875,  e  =  0.21  +  0.25  +  0  =  0.46 


-.1584- 


.947  -  1.875  -  .46 
.158  -  .314  =  -  0.156 
(in  front  of  first  lens) 


COMBINATIONS   OF   LENSES  79 

A'N'  +  31'  =  -  0.0625  +  ~  M  X.(~fi  L875)  - 

-  I.ooo 

-  0.0625  -  0.621  =  -  0.6835 
(in  first  lens) 

7.   Three  lenses,  in  contact.     Light  from  left. 
First  lens,     r  =  -  4.29,  s  =  -  1.20,  /*  =  1.5146,  e  =  .230 
whence/  =  --  12.627,  /'  =  3.532,^  =  3.157,  A  N  =  .2056, 

A'N'  =  0.0573. 
Second  lens,     r  =  -  1.20,  s  =  -  3.75,  /*  =  1.574, 

e  =  0.050 
Whence/  =  -  3.2906,  /'  =  10.2831,  F  =  -  3.094, 

AN  =  -  0.015,  A'N'  =  -  0.047. 

Third  lens,     r  =  -  3.75,  s  =  -  1.8,  /*  =  1.517,  e  =  0.151 
Whence  /  =  --  11.0033,  /'  =  5.2814,  F  =  6.528, 

AN  =  0.186,  A'N'  =  0.089 

Combination  of  1st  and  2d  lens,    /i  =  3.157,   /2  =  —  3.094, 

e  =  -  0.0723 

™  3.157  (-  3.094) 

Whence  F  =  Q  =    -          =     -71.76 


0.2056  +- 

0.2056  -  1.68  =  -  1.4744 
A'#'  +  ft'  =  -  0.047  -  1.65  =  -  1.697 

Notice  that  e  is  negative  because  the  N'  of  the  first  lens 
and  the  N  of  the  second  lens  are  crossed. 
First  combination,  combined  with  3d  lens. 

/i  =  -  71.76,  /2  =  6.528,  e  =  1.697  +  0.186  =  1.883 

-  71.76  X  6.528 


Whence  F  = 


-  71.76  +  6.528  -  1.883 


80 


THICK-LENS    OPTICS 


-  468.4492  _ 
-  67.115 

1.883  (-  71.76) 


-  67.115 
-  1.474  +  2.013  =  0.539 


A'N'  +  W  =  0.089  + 


0.089  +  0.186  =  0.275 
8.    Three  doublets  combined,  see  diagram. 


Line  of  vertices 


...i-\ .......;.-..._ ...-^.j  ^4      Line  of  lens  nodals 

jvn  \'        Line  of  doublet  nodals 


k-  ......  2.00---  -*• 


.  . 

.......  -2.77---'-  -------  *• 

—  3.96-- 


-•/.04-M 


Nodals  of  first  combi- 
nation 

Nodals  of  second  com- 
bination 


1  8 

First  lens  of  1st  doublet.      r  =  GO,  s  =  1,  e  =  -,//,  =  - 

—  o 


Whence 


i.  oo 


-  1 


00 


L_8_l        8    1  +  0  -  0       16 

3        2 


A'N'  = 


Q  ,  '  '_f  1     8\ 
8  (  oo  +  /  -  e) 


=  0 


-8 


8'oo-|-i       8*1-0-0 


COMBINATIONS    OF    LENSES  81 

3 

Second  lens  of  1st  doublet,     r  =  I,  s  =  —  1,  e  =  1,  /A  =  - 

2 

Whence         /  =     •  2  -  1  =3 


2-3-3 
r   =  u- 


A'N'  = 


3  (3  +  3  -  1)       5 

2-1-3  2 

3  (3  +  3  -  1)  ~~  5 

-1-3-2         _  2 
3-5  5 


562 
VsJ  doublet.    f\  =  —  -,  /2  =  r-,  c  =  - 
o  5  o 


=  2-3077 


^'  +  9J'=-     +         r-=  -     +       =  0.15385 

2  8 
First  lens  of  2d  doublet,     r  =  oos  =  4,  e  =  ->«,  =  - 

3  5 

Q 

Whence         /  =  -  •  oo  =   oo 
5 

,  84  32 

;          ~5*3'  ~T 

5  .  <x>  (-  3f)  -  5  •  32  =  -  20 

8  (oo  -3^2  -|)  3.  8  3 

A  N  =  -    -  QQ  =  2    5       _5_ 

5  '  8  '  "oo  -  V  -  §  ~  3  '  8  ~  12 


82  THICK-LENS   OPTICS 

15        -  32 


A  '  N'  — 


oo 


4  3 

Second  lens  of  2d  doublet,     r  =  4,  s  =  —  4,  e  =  ~>  /u.  =  - 

o  .Z 

Whence        /  =  |  -  1  •  2  =  12 
z    i 


2  12  -12  72 
"  3  '  12  +  12  -  I  "  17 

42    12-3       J*_ 

3  '  3  '     68      =  17 

A'N'  -     ~4    2    12     3  8 

~3~  '  3  '       '  68  "     "  T7 

20-72  8 

Second  doublet,    fi  =      -  -^  >  /  2  =  jy  >  €  ==  — 

-  90     79  1 

Whence        F  =  -f°  -  g  •  ^     ^    -g  =  9.73 

3       "17       17 

A    ~20 

A  AT   .   c»        5    .  17'     3  5        40 

A  AT  +  91  ==  12+  -  20   .   72    "8"  =  12  +  37  =  L498 
3       h  17       17 

A'  N'  +  ft'  =  ^  +  ^y^l  =  -  .4705  +  .  6868  =  0.2163 

3  8 

First  lens  of  3d  doublet,     r  =  <x>,s  =  10,  e  =  ±>  P  =  § 

-  8    10  -  80 

Whence         /  =  oo,/r  -  —  -       -  5  =  —   - 


COMBINATIONS   OF   LENSES                         83 

F       5 

-  80 

1                 -5-10 

8 

3 

c 

80       3             3 

50 

3       4 

3 

AN       3 

5 

8' 

00 

15 

j±  i\  —   .  • 
4 

00    —    .    •    • 

32 

4       8        3         oo 

3  3 

Second  lens  of  3d  doublet,     r  =  10,  s  =  —  10,  e  =  ~  ,  /*  =Q 

O         If) 

Whence        /  =  £  .  ~  .  2  =  30 
z     i 

-3      -10 

f    "2     r 

2  30-30  400 

~  3  '30  +  30  -f  ==   39  • 

3  2  30  _  20 
=  2  '  3  '  30  +  30  -  f  ~  39 

-  3    2    30    2       -  20 


A    IV 

Tfa'rd  doublet. 
Whence        F 

A  \T  4-  9f> 

2 

/    -  - 

3      117 
-  50 

39 
400 

20 

.400 
81 

4- 

Ji          3     '  •/2 
-  50    400 

39' 

1 

39 
5 

3 

24 

39      - 

.6923 
-  20 

50 
3 

400 

20 
39 

=  0  4R87S  - 

39 
50 

39 

3 

_ 

"3         39        39 
1.23456  =  1.70331 


84  THICK-LENS    OPTICS 

-  30 
A'N'  +  9fe'  -  -^  +  0.75975  =  -  0.51282  + 

oU 

0.75975  =  0.24693 

Combination  of  second  and  third  doublet,  so  that  the  posterior 
nodal    point    of   the    second    doublet    coincides  with    the 
anterior  face  of  the  third  doublet. 
This  makes  *  =  1.70331,  /i  =  9.73,  /2  =  24.6923 

9.73  X  24.6923  240.253 


9.73  +  24.6923  -  1.70331       32.7190 
7.343 


Whence       F  = 


A  N  +  ft  =  1.498  +         327'      =  L498  + 

1.277  =  2.775 
'*  +  V  =  0.24693  +  " 


0.247  -  1.285  =  --  1.038 

Combination  of  first  doublet  with  preceding  combination,  so 
taken  that  the  posterior  nodal  plane  of  the  doublet  coin- 
cides with  the  anterior  face  of  the  second  doublet. 
This  makes  c  =  2.775,  /i  =  2.3077,  /2  =  7.343 

„  2.3077  X  7.343  16.948 


2.3077  +  7.343  -  2.775 

AN  +  N  =  1.08173  +  2.3077X+2'3077  =  L082 
=  1.082  +  .918  =  2.000 


A'N'  +  W  =  -  1.038  +  '        =  -  1.038  - 

=  -  1.038  -  2.921  =  -  3.959 


COMBINATIONS    OF    LENSES  85 

9.    Camera  lens,  composed  of  three  lenses,  light  from  left, 
First  lens.     ^  =  1.6103,  r  =  1.264,  s  =  1.48,  e  =  0.105, 
air  space  =  0.232. 

Second    lens.     /*  =  1.61(|3,     r  =  -  2.09,     s  =  -  0.553, 
e  =  0.358,  air  space  =  0.0053. 

Third      lens,     n  =  1.524,      r  =  -  0.5325,      s  =  -  2.8, 
e  =  0.110. 

,  .        1.6103  X  1.264       0  OOC1 

/ens.    /i  =  --  o~6103  --  =  3'3351 

-  1.6103  X  1.48 
0.6103 

_  -  3.3351  X  3.9050 
"  1.6103  X  (-  0.6749)  " 

3.3351  X  0.105 
"1.6103  (-0.6749)  ' 

_  -  (-  3.9050)  X  1.05  _ 
1.6103  (-  0.6749) 


Second  lens. 

1.6103  X  (-  2.09) 
/1=  0.6103 


-1.6103  (-0.553) 
0.6103 

-  5.5145  X  1.4591 
1.6103  (-4.4134) 

T       -  5.5145  X  0.358 

=°'2778 


1.6103  (-  4.4134) 

_  -  1.4591  X  0.358  _ 
~  1.6103  (-4.4134)  ~ 
Third  lens. 

=  1.524  (-  0.5325)  _  _ 
•"  0.524  1' 


86  THICK-LENS   OPTICS 

-  1.524  (-  2.8) 
/2=  0.524 

-  1.5487  X  8.1435 
1.524  X  6.4848 

.  ,.       -  1.5487  X  0.110  nm7Q 

AJV=       1.524  X  6.4848  0'°173 

-  8.1435  X  0.110  _     _009064 
1.524  X  6.4848 

First  combination,  of  2d  and  3d  lens. 

/!  =  1.1322,  /2  =  -  1.2762,  c  =  -  0.0854 
AN  =  0.2778,  A'N'  =  -  0.0906 

Therefore     A  N  +  %  -  0.2778  + 


0.2778  +  1.6499  =  1.9277 


A'N'  +  V  =  -  0.0906  +  ~  <~  **W<-  a°854>  = 

—  u.uooo 

-  0.0906  +  1.8598  =  1.7692 
1.1322  (-  1.2762) 

-  0.0586 
Second  combination,  of  1st  lens  and  1st  combination. 

/i  =  11.9837,  /2  =  24.6561,  €  =  2.5369 
AN  =  -  0.3222,  A'N'  =  1.7692 

Therefore     A  N  +  *  -  -  0.3222  +  1L98! 


-  0.3222  +  0.8915  =  0.5693 
A'N'  +  »'  =  1.7692  +  - 


1.7692  -  1.8345  =  -  0.0653 
11.9837  X  24.6561 


34.1029 


=  8.664 


COMBINATIONS    OF    LENSES 


87 


Look  out  for  negative  e, 
To  calculate  e,  locate  the 


Excellent  for  care  in  signs, 
caused  by  the  crossed  jiodals. 
nodals  on  a  rough  diagram. 

108.   Magnifying  Power  of  a  Microscope  (Compound). 


Magnifying  power  =  -  •  -  =  1st  magnif.  X  2d  magnif. 
o     % 


(§55) 


/  =  focal  length  of  the  eyepiece 
D  =  least  distance  of  distinct  vision 
v  =  dist.  from  objective  to  image 
u  =  dist.  from  objective  to  object 

0  =  size  of  object 

1  =  size  of  real  image 

7  =  size  of  virtual  image 

109.  N.  B.  —  In  the  microscope  the  magnifying  power  is 
the  ratio  between  the  virtual  image  and  the  object,  because 
both  are  seen  at  the  same  distance,  the  distance  of  distinct 
vision.  In  the  telescope,  however  (see  next  section),  the 
virtual  image  and  object  are  not  seen  at  the  same  distance 
and  the  comparison  must  be  made  on  a  different  basis; 
viz.,  comparison  of  the  angles  under  which  the  virtual 
image  and  the  object  are  seen.  The  distance  at  which 
the  virtual  image  is  to  be  considered  depends  upon  the 
"  set  "  of  the  eye  of  the  observer.  A  person  with  a  very 
flexible  eye  can  vary  the  distance  from  far  to  near,  which 


88  THICK-LENS    OPTICS 

produces  a  .slight  variation  in  the  angle.  A  far-sighted 
or  presbyopic  eye  has  the  eye  "  set  "  for  the  far  distance, 
and  therefore  for  the  slightly  smaller  angle  subtended  by 
the  virtual  image,  which,  however,  is  practically  the  same 
angle.  Ratio  of  angles  could  have  been  used  in  the  micro- 
scope also. 

110.  Magnifying  Power  of  a  Telescope. 


Angle  under  which  object  would  be  seen  by  the  naked 
eye  =  a,  practically. 

Angle  under  which  object  would  be  seen  by  the  tele- 
scope =  /3. 

O  Jj» 

Therefore,  magnification  =  -  =  —  approximately,  since 
the  angles  are  small. 

BU.836.  -          =  o-     1-B-' 


F       F    D  +f  F 

and  -  =  y      D        =  mag.  =  y  nearly 


For  the  eye  looking  at  a  landscape,  —      -  is  approximately 

F 

1,  and  the  magnification  =  j 

=  distance  of  distinct  landscape  vision 

=  focal  length  of  the  eyepiece 

=  focal  length  of  the  object  of  glass 

Example.  —  Object  glass  of  telescope  is  20ft.  focal  length. 
With  a  \  inch  eyepiece,  what  is  the  magnification? 
Ans.     Mag.  =  480. 


COMBINATIONS    OF    LENSES 


89 


111.   Magnifying  Power  of  Opera  Glass  (Galilean  tele- 
scope). 

a  =  angle  of  object  at  eye  of  observer,  practically 
ft  =  angle  of  image 
D  =  least  distance  of  distinct 

vision 

F  =  focal  length  of  object  glass 
/  =  focal  length  of  eyepiece 

Magnification  =  -  (practically  exact  for  distant  objects) 


D  —  f      F 

—          --      nearly 


-•» 


] 


Example.  —  If  the  object  glass  is  4  in.  focus,  and  the  eye- 
piece 1|  in.,  what  will  be  the  magnifying  power  and  the 
distance  between  the  lenses? 

Ans.     8/3;    2.5. 

112.   Magnifying  Power  of  Camera.  —  (See  §§  47,  53.) 

For  telephoto  camera,  see  §  120. 


CHAPTER  V 

TELEPHOTO    LENS 

113.  In  §  29  we  found  that  a  negative  lens  interposed 
in  the  path  of  converging  rays  so  that  the  aerial  object 
fell  within  the  focal  distance  of  the  negative  lens  gave  a 
real  image  beyond  the  aerial  object. 

Jl 

t 


»  v%^       JIUVK  fwul  distance 


— •  Equivalent  focal 

This  is  the  principle  of  the  telephoto  lens,  the  aerial 
object  being  the  real  image  made  by  the  camera  lens. 
As  shown  in  the  diagram,  the  result  of  interposing  the 
negative  lens  is  to  give  an  image  as  if  made  by  a  long 
focus  lens  in  the  position  A.  But  a  long  focus  lens  gives 
a  large  image,  and  usually  requires  a  long  camera  box; 
i.e.  the  long  back  focal  distance.  By  reference  to  the 
diagram  it  will  be  seen,  however,  that  in  the  case  of  the 
telephoto  lens,  the  back  focal  distance  is  very  much  less 
than  the  focal  distance.  Hence  instead  of  a  very  long 
focus  lens  with  its  correspondingly  long  box,  we  have  the 
combination  of  two  lenses,  a  -f  and  a  — ,  and  the  same 
effect  with  a  very  much  shorter  box.  See  §  107,  Ex.  2, 

§  93,  Ex.  7. 

90 


TELEPHOTO    LENS  91 

114.  Focal  Length  of  a  Telephoto  Combination. 

By  §§  94,  95,  focal  length  =  F  =  ,  ~  ^  '  f 2 

Jl    —    12    —    € 

back  focal  distance  =         2    ^ —  =  FB 

Ji  -  12  -  « 

/i  =  focal  length  of  +  lens 
/2  =  of  2d  lens 

e  =  distance  between  the  nodes,  emergence  of  1st  lens, 
incidence  of  2d  lens 

115.  Telephoto  Magnification. 

90?  =  increase  of  magnification  due  to  combination  as 
compared  with  the  +  lens  alone 
size  of  image  made  by  combination 
size  of  image  made  by  converging  lens 
focal  length  of  combination 
focal  length  of  converging  lens 

I  /'/•  /if 2    ,  -fa 

"  /I  (/I   +  /*    -    «)/!  (/I   -    *2   -   «)/!         /  -    fj  ~  « 


i  i  =          )  i 

, 


2  (/I    —    f  2    —    €)  f 

back  focal  distance 


_ 

num.  val.  of  focal  length  of  neg.  lens 
Therefore  FB  =  back  focal  distance  =  f2  (2ft  —  1) 
F  =  equivalent  focus  of  combination 


2 

Notice  that  FB  for  a  given  2ft  is  affected  only  by  the 
negative  lens  used. 

EXAMPLES 

1.  For  m  =  3,/2  =  -  3,  we  get  FB  =  3  (3  -  1)  =  6. 

2.  Rays  forming  a  real  image  are  intercepted  by  a  con- 


92  THICK-LENS    OPTICS 

cave  lens  of  12  in.  focal  length  at  a  distance  8  in.  from  the 
screen.  How  far  must  the  screen  be  moved  to  be  in  focus 
again? 

Ans-     Tt  =  I  ~  ~    /.  »  =  24,  /.  24  -  8  =  16  =  distance 
LA       o        V 

to  be  moved. 

3.  For  /i  =  6,/2  =  -3, 

1.  If  FB  =  12     then  2ft  =  5      F  =  30. 

2.  If  <m  =  3}     then  FB  =  7J     F  =  21. 

3.  If  FB  =  7}     then  F    =  21. 

4.  Dallmeyer's  Telephotographic  lens,    /i  =  6,  /2  =  —3. 


Since  F  must  be  +,  e  >  3,  say  3|. 

Hence  AW  =  -  30,  A'W  =  -  15,  F  =  24,  FB  =  9. 

5.  In  Ex.  5,  suppose  FB  =  12,  what  should  e  be,  and 
what  will  be  the  value  of  F? 

Ans.     e  =  3f  ,  F  =  30. 

6.  /i  =  9J,  /2  =  --  13. 

Ans.     AW  =  -  5,V,  AW  =  -  6H,  F  =  16&,  FB  =  9A- 

7.  /i  =  --  13,  /2  =  9|. 
Ans. 


Notice  that  this  is  the  lens  of  Ex.  6  turned  around,  and 
observe  the  large  increase  of  FB  when  the  negative  lens  is 
in  front. 

116.  Focal  Distances. 

fA  fA       l/i2          fs2 

r    =  -; ^ =    — r-    = 7   =   m  T1 


/i  —  /2  — 


TELEPHOTO    LENS 


FNi~ 

4/*_^ 
d         d 


I  Ffi  means  the  distance 
|_  between  F  and  /i,  etc. 


_= 

d       ~  d 

=  mp  =  distance  of  front  focal  point  of  the  combina- 
tion from  the  front  focal  point  of  the 
positive  lens 

(—  A  Ni,  because  ANi  is  essentially  negative,  and  FNi  essen- 
tially positive,  and  the  —  sign  is  needed  in  order  to  give 
them  the  same  combinative  sense) 


Similarly  ^j-  =  —  =  distance  of  back  focal  point  of  the 
combination  from   the  back  focal 
point  of  the  negative  lens 
Hence     Distance  between  front  focal  point  of  the  system 

and  the  front  lens  (nodal  point)  =  mF  +  /i 
Distance  between  back  focal  point  of  the  system 

F  F 

and  the  negative  lens  = h  /2  = $2 

m  m 

117.    For  a  single  lens,  §  47  " 

v  =—=+/=  dist.  to  image 


THICK-LENS   OPTICS 

u  =  -  (Nf  +  /)  =  dist.  to  object 

•• 

-^r  =  magnification  =  -  —  1 

N  =  reduction  factor 
Similarly  for  the  telephoto  combination 

/  =  dist.  of  image  from  neg.  lens  (nodal  point) 
F    ,   F 


=  N  +  m-f*  §116 

0  =  dist.  of  object  from  pos.  lens  (nodal  point) 

-(NF  +  mF+fi)  §116 

N  =  reduction  factor  for  the  system 
F  =  focal  length  of  the  system 

=  focal  length  of  +  lens  • 
/2  =  focal  length  of  —  lens 


-f,--/, 

E  =  bellows  extension 

Example,  —fi  =  6,  /2  =  -  3,   e  =  3J.     The  results  are 
given  in  the  diagram. 


118.   Focal  Radius  in  Terms  of  the  Magnification, 


N 


[§47 


Therefore 


•-GH 


TELEPHOTO    LENS  95 

But     v  -  F  =  E  -  FB  (See  diagram) 


therefore  FB  =  E  -  -  -^ 


But 


F  =  m  FB  +  /i 


[§115 


whence      F  = 


mE+fi 


£J  =  bellows  extension 

/i  =  focal  length  of  +  lens 

N  =  reduction  factor 

If  N  =  oo  (i.e.  object  very  distant),  this  becomes 
F  =  mFs  +  /i,  as  before 


EXAMPLES 

1.  /i  -  10,  /2  =  -  4,   with  the   object  80  distant,   to 
reduce  to  J  size. 

Mag.  produced  by  1st  lens  =    Q  _  1Q  =  ^  [See  §  47 

.'.     M  =mag.  of  1st  image  =  |  [M  •  |  =  J 

...      #  =  4  (£  -  1)  =  10  [§  47 

V>  -10  +  10  = 

2.  If /i  =  8,  /2  =  -  3,  and  object  is  60  in  front,  N  =  2. 

1st  mag.  =  y2^,  M  =  J3p-,  FB  =  5iS  ^  = 


96 


THICK-LENS    OPTICS 


3.  /i  =  10,  /2  =  -  4,  with  the  object  167J  in  front  of 
the  +  lens,  to  reduce  to  J  size. 

M  =  <%3         E  =  27J        F  =  35 
Other  facts  are  shown  in  the  diagram  below. 


•flO 


JV, 


A'W  =  -  25     F  =  35     FB  =  10 
u  =  -  105  v  =  52J 

4.    Positive  lens  of  8  in.  focus  and  negative  lens  of  3  in. 
focus  with  the  object  60  in.  away,  to  reduce  to  |  size. 
Mag.  of  +  lens  =  TV,  M  =  \3-,  E  =  V,  F  =  V- 

119.   Distance  to  Object  for  a  given  Magnification,  ==  • 

M  =  magnif.  of  1st  image  due  to  2d  lens 
final  image       E  +  f 2 


1st  image  f2 

rr 

#    =  extension  of  bellows  =  FB  +  -TT 


(Conf .  §  47) 
[§  118 


TELEPHOTO    LENS  97 

FB  =  F-^±  [§  115 

-  •  M  =  -~  =  1st  reduction  X  mag.  due  to  2d  lens  =  final 
reduction 

u    =  dist.  of  object  =  fi  (n  +  1)  fi  A 

L»  '  r=7i'  5  47 

Find  FB,  then  E,  then  M,  then  n,  then  M. 
Example.  —  F  =  24,  /!  =  6,  /2  =  -  3,  N  =  5 
.-.     FB  =  9,  £J  =  13|,  M  =  5J,  n  =  28 
.-.       w  =  6  (28  +  1)  =  174  in. 
120.   Reduction  Factor. 


From  §  118   N  =      „  _T  --  =j.     For  TV,  see  §§  119,  117. 
~r  Ji  —  " 


CHAPTER  VI 
REFLECTION    AT    SPHERICAL    SURFACES 

NOTE.  —  This  chapter  is  introduced  on  account  of  some  experi- 
mental observations. 

121.   The  angles  of  incidence  and  reflection  are  equal.  (See 

any  text-book  on  physics.) 

By  geometry,  a  line  bisect- 
ing the  angle  of  a  triangle 
divides  the  opposite  side  into 

u ._ _. ^   segments  proportional  to  the 

adjacent  sides;  hence,  since 

the  angle  of  reflection  is  equal  to  the  angle  of  incidence 

a  _  c 
b~d 

But  for  points  near  the  axis,  d  =  u,  c  =  v,  whence 

a  v  r  —  v  v 
r  =  -»  or  -  -  =  - 
o  u  u  —  r  u 

1       1       2 

or  -  +  -  =  - 

v       u      r 

For  u  =  oo,  i.e.  parallel  rays  from  a  distant  object, 
Vf  =  2  =  ft  called  the  focal  distance,  whence 

1       1  _  2  _  1 
v       u      r       f 

This  is  a  general  equation,  applicable  to  convex  or  con- 
cave surfaces,  attention  being  paid  to  the  values  of  the 
letters,  measurements  to  the  right  being  positive. 

98 


REFLECTION    AT    SPHERICAL    SURFACES 


99 


122.  In  reflection,  it  simplifies  the  questions  of  signs 
if  we  suppose  the  light  to  come  from  the  right,  thus  making 
u  positive  for  real  objects  and  negative  for  aerial  objects, 
with  the  following  tabulation  of  results. 


Divergent  pencil 
Real  object 


Convergent  pencil 
Aerial  object 


concave  mrror 


convex  mirror 
concave  mirror 


convex  mrror 


>2 


Negative  v  means  the  image  is  virtual. 

123.    Graphic    Construction   for   the   object   in    various 
positions;   determined  by  known  lines,  focal  and  central. 


.Aerial  Urnaae^     Object 

object  Focus    * 

The  order  of  the  letters  denotes  the  order  of  construction. 


Caution.  —  This  is  accurate  only  near  the  vertex,  and 
is  introduced  here  in  order  to  illustrate  the  method.  In 
actual  construction,  symbolize  the  surface  by  a  straight 
line  perpendicular  to  the  axis,  as  was  done  in  the  case  of 


100  THICK-LENS    OPTICS 

a  thin  lens.     For  tracing  any  ray,  we  have  similarly  to 
§  32,  rs  —  >  <j>  \  \  to  c,  where  s  =  surface,  c  =  line  through 
center  of  curvature. 
124.   Magnification. 

TV/T       -c     4.-  Image       v  —  r       v 

Magnification  =  ^,  .    -  =  -       -  =  - 

Object       r  —  u      u 

EXAMPLES 

1.  A  candle  flame  1  cm.  long,  36  cm.  in  front  of  a  concave 
mirror,  whose  focal  length  is  30  cm.,  gives  what  image  and 
where? 


Ans-       +       ^         .'.•'  =  180.    Mag.  =         =  5 

2.  A  flame  2  in.  in  front  of  a  positive  lens  of  1  in.  focus, 
and  plane  mirror  }  in.  behind  the  lens,  reflects  the  rays  back 
through  the  lens.     Show  that  the  real  image  will  be  J  in. 
from  the  lens.     First  image  is  2  behind  the  lens  or  1J  be- 
hind the  mirror.     Second  image  (aerial)  1  J  before  the  mirror 
or  1  before  the  lens.     Third  image  is  J  before  the  lens. 

3.  How  far  from  a  concave  mirror  must  an  object  be 
placed  to  be  magnified  n  times? 

Ans.     v  =  nu  (real  image);    v  =  —  nu  (virtual  image). 


11        1  (n 

For  real  image,  -?  =  -  H  ---     .'.  u  =  -  -  •     For  vir- 

'  /       v       nu  n 

11  1  (n-l)f 

tual  image,  -.  =  —  —    —     .'.  v  =  -  —  • 

f      u       —  nu  n 

4.    Gas  flame  10  in.  from  wall.     Required  real  image  on 
the  wall  three  times  as  large.     What  mirror  and  where? 
Ans.     v  =  3  u'  u  =  dist.  from  mirror  to  object;  10  -f  u  = 

3u.     .'.  u  =  5.     -7  =  -=  +  ^  -  .'.  f  =  3f  .     Result,  a  concave 
j        5       15 

mirror,  3f  focal  length,  5  in.  from  object. 


REFLECTION    AT    SPHERICAL   SURFACES 

5.  u  =  10,  /  =  -  30. 

Ans.     v  =  -  *£-.     Mag.  =  -  f . 

6.  Object  =  1  in.,     u  =  18,  /  =  +  15. 
Ans.     v  =  90.     Mag.  =  5. 

7.  u  =  if. 

Ans.     v  =  —  /.     Image  virtual  =  twice  object 

8.  Mag.  =  12.     Object  11  from  screen. 
Ans.     12  u  —  u  =  11.     /.  u  =  1,  .'.  /  =  if. 


101 


CHAPTER  VII 
EXPERIMENTAL    OBSERVATIONS 

To  get  thoroughly  satisfactory  results  requires  care, 
experience,  and  a  trained  eye.  The  average  untrained  eye 
cannot  see  things  as  they  actually  exist.  Make  several 
observations  and  take  the  mean  of  them. 

125.  To  Find  Radius  of  Curvature  of  a  Surface. 


Telescope  \swface 

—  "<=L^z  -------  \-  ------  •  —  .....  -----  —  -fk 

j*  .......  _  ....................  ^  .........  _„  ........  _______  J\ 

Scal 


The  lights  (or  other  suitable  objects)  produce  two  images, 
the  distance  apart  of  which  is  observed  through  the  tele- 
scope on  the  scale. 

2  Al 

r  =  radius  of  curv.  =  7  --  x^  (for  convex  surface) 


L  -2 
2AI 


(for  concave  surface) 


L  +  21 

I  =  distance  of  the  images  apart 
L  =  actual  dist.  of  the  lights 
A  =  dist.  from  surface  to  line  of  the  lights 

The  less  the  curvature  the  greater  A  must  be  to  get 
accurate  results.  In  the  case  of  biconcave  or  biconvex 
lenses  it  is  easily  seen  which  images  are  from  the  front  or 
back  surface,  by  their  inverted  or  erect  position. 

102 


EXPERIMENTAL   OBSERVATIONS  103 

Proof.  —  The  distance  a,  at  which  the  virtual  image  of 
the  line  L  is  formed  behind  the  surface,  is  given  by  (see 

§  121). 

11,2  Ar 

-  =  -T-  +  -  >  whence  a  =  =-T  —  ;  — 
a      A      r  2  A  -f-  r 

X      r  —  a     1  A 


L      A  +  r    A      A  -{- a 

1  r 

whence  =-  = 


[A  =  length  of  image 


L      2  (A  +  r) 

2A1 

whence  r  =  = ^-r 

Similarly  for  a  concave  surface. 

(Second  method.)  Make  object  (small  illuminated  disc 
in  screen)  and  image  coincide  by  reflection  from  concave 
mirror,  or  by  interposing  a  positive  lens  in  front  of  the  con- 
vex mirror. 

For  concave  mirror,  r  =  2/  equals  the  distance  of  the 
mirror  surface  from  the  image  (object). 

For  convex  mirror,  find  image  of  disc  when  mirror  is  re- 
moved. The  distance  of  this  point  from  the  front  of  the 
mirror  before  removal  is  r. 

If  object  and  image  do  not  coincide,  /  can  be  calculated 
from  the  u  and  v  distances.  Check  graphically  by  diagram 
1  of  §  40. 

Or  for  a  concave  mirror  /  can  be  measured  directly  by 
first  rendering  the  rays  parallel  by  a  positive  lens;  the  dis- 
tance of  the  image  from  the  mirror  is  /. 

126.  By  Matching  with  Surface  of  Known  Curvature, 
using  the  scale  on  both  as  in  §  125. 

This  matching  can  be  roughly  done  by  holding  the  sur- 
faces in  the  hand,  and  observing  the  image  of  some  bright 
object,  window,  lamp  globe,  etc. 


104  THICK-LENS    OPTICS 

127.   Radius  of  Curvature  of  Surface  of  Small  Curvature. 

Focus  a  telescope  on  a  scale  at  a  distance  A  from  the 
object  glass.  With  the  telescope  thus  focussed,  let  the 
image  of  an  object  reflected  from  the  surface  to  be  tested 
be  clearly  seen  when  the  distance  between  the  object  and 
the  surface  is  a,  and  that  between  the  surface  and  the 
object  glass  of  the  telescope  is  e.  Then 

A  -  e 


rad.  of  curv.  =  r  =  2  a 


T->      f      ^       v  —  e  -\-  A  ^     T .  ,    . 

Proof,     j-  = — = ,  L'  being  practi- 

2 

cally  at  the  surface  [Sim.  triangles 

Whence  r  =  2  a  -: — 

A  —  e  —  a    ) 

The  best  value  for  e  is  about  -~  • 

Positive  r  denotes  a  concave  surface ;  negative  r  a  convex 
surface. 

The  absence  of  parallax  between  the  cross  wires  of  the 
telescope  and  the  image  is  the  test  of  distinct  vision. 

FOCAL    LENGTH    OF    A    THIN    POSITIVE    LENS 

128.  With  the  Sun.  —  Distance  of  the  image  of  the  sun 
from  the  lens  =  /. 


EXPERIMENTAL    OBSERVATIONS  105 

129.  Lens  Distances.—  /=-^-.  (See  §  16.)    Distances 

u      v 

to  the  left  are  negative,  u  =  distance  from  lens  to  object, 
v  =  distance  to  image.  For  accurate  focussing,  see  §  156. 
Solve  graphically  by  diagram  1  of  §  40,  for  check. 

130.  With  a  Telescope.  —  Focus  the  telescope  on  some 
distant  object.     Place  the  lens  in  front  of  the  object  glass. 
Look  through  the  telescope,  without  altering  its  length, 
at  some  plane  object  (a  newspaper),  adjusting  the  distance 
for  distinct  vision.     The  distance  of  the  object  from  the 
lens  =  /,   because  the   lens  sends  parallel  rays  into  the 
telescope  already  set  for  parallel  rays. 

131.  By  Different  Positions  of  the  Lens. 

/  =  -  u  — pi .     I  =  distance  between  image  and  object, 

a  =  distance  between  the  two  positions  of  the  lens  when 
giving  a  distinct  image,  the  object  and  screen  remaining 
fixed. 

Proof.  —  The  distances  of  object  and  image  from  the  lens 
are  \  (I  +  a)  and  \  (I  —  a),  whence  (§  17), 

1          22  41 


/      1  -  a   '  1  +  a      I2  -  a2 

132.  From  Equality  of  Object  and  Image.  —  Distance 
between  object  and  image  =  4  jf.     (See  §  37,  Ex.  2.) 

133.  Comparison  of  Images.  —  A  candle  (or  illuminated 
aperture)  is  placed  a  distance  a  from  a  screen  and  the 
image  focussed  on  the  screen.     On  moving  the  lens  towards 
the  candle  another  image  is  formed  which  is  m  times  as 
large  as  the  former. 

The  focal  length    = 


(1  + 


106  THICK-LENS    OPTICS 

Proof:    By  §  131,  /  =  °-^~ 


\b  =  distance  between 
2  positions  of  the  lens 


-  1 


2/ 
From  the  1st  eq.     62  =  a2  —  4  a/ 

From  the  2d  eq.       6  =  (a  ~ 


m  +  1 

(a2-4a/+4/2)  (m  -  I)2 


(m  +  I)2 

(g2-4a/+4/2)(m-l)2 
(m  +  1)2 

ma2 


(m  -  I)2 

—  2  am 

(m  -  I)2 


/     ma2  4  a2m2 

VI^ny^Tm"--  I)4 

M 


—  2am  a         /       ,      4m2      _  m  (m2  +  2m  +  1) 

(m  -  l^  +  m^T  V^    ^(m-l)2" 

—  2  am  +  a  (m  + 


(m  -  I)2 

a  Vm  (m  +  1  -  2  Vm)  =  /-  /Vm  -  IV 

(m  -  I)2  \m-iy 

Vm  —  1  \2             a  Vm 


(Vm  -  1 

Q.E.D. 


EXPERIMENTAL    OBSERVATIONS  107 

FOCAL    LENGTH    OF    THICK    POSITIVE    LENS 
134.   From  Highly  Magnified  Image. 


I  =  length  of  a  division  of  the  scale, 
L  =  length  of  the  image. 

Proof:    1  +  1  =  1.    (See  §  16.)     ^  =  -  •     Whence/  = 

V         U         J  l          U 

v  j  —  —  -;  •     Since  v  is  very  large,  small  errors  in  its  measure- 
L  +  I 

ment  or  mistakes  in  locating  the  nodal  point  (to  which  it 
should  be  measured)  do  not  materially  affect  the  result. 

.Lens 


White  screen  - 

wttfiyreatly  magnified  dlumlnatefL  scale 

image  *"  (onj?lass)~ 

Screen  and  scale  may  be  interchanged,  with  diminished 
image,  using  a  lens  to  read  the  image,  in  which  case 

L 

135.   Swing  of   Camera  or  Lens   Carrier.  —  Swing  the 
camera  or  lens  carrier  horizontally  on  a 


table  (guide  by  a  flat  stick  with  a  small 

nail  through  one  end)  until  a  distant  ver-  c 

tical  object  is  focussed  on  two  vertical  \'/ 

lines  (short)  on  the  extreme  edges  of  the  :'/ 

ground  glass  screen.     Mark  the  angle  of  // 

swing  on  the  table.     In  the  angle  plot  c/ 

the  distance  a,  between  the  lines  on  the   4    °  =  distance  be- 

'        .  tween    lines    on  the 

screen,  perpendicular  to  the  bisector  of   gcreen>  &=fOCaldis- 

the  angle.     The  bisector,  6,  will  equal   tance 
the  focal  length. 


108  THICK-LENS    OPTICS 

136.   Movement   of   Screen.  —  Focus   on   very   distant 
object.     Then  focus  on  a  near  object,  making  the  image 


and  object  equal  (same  size  discs  with  parallel  lines,  one 
pattern  covering  the  other).  /  =  the  distance  the  screen 
is  moved.  (Conf.  §  37,  Ex:  2;  §  47,  Ex.  15.) 

137.   Movement  of   Screen.  —  Focus   for   very   distant 

object.     Focus  on  a  near  object  for  the  image  =  -=  of  object. 

(s  =  number  of  units  on  scale,  d  =  number  of  units  on 
screen  covered  by  the  s  units  of  the  scale.) 


Th                f  -  —  I     =  distance  moved  by  screen 

d  [between  the  two  focussings 

Proof:         f  =       V  \~v  =  dist- to  imaSe-     See  § 47 

m  +  1  [m  =  magnification 

B  /. 


m  +  1 

TTT,  .      a       sa  ,  .,       d 

Whence  /  =  —  =  -y  [m  =  magmf .  =  - 

m       d  s 

138.   Angle  of  Vision.    /  =- (See  §  2.)     h  can  be 

tan  ct 

Hstant  object 


Distant  object 


measured  by  a  scale  or  a  sliding  lens,     a  must  be  measured 
by  an  instrument,  or  tan  a  can  be  found  by  §  157. 
139.   Unit  Screen  Movement  (Lionel  Laurence) 


/  dmn 
\/m  _  n  = 


if  m  =  2n,  n  =  1 


EXPERIMENTAL    OBSERVATIONS 


109 


D     f  !  !  !       i          K      f(f  +  n)       . 

Proo/:      f-—  +  fc  +  a  =  r  whence  b  =  -  >—l  -  d 

[§74 

h  "TT  -f       B 

zi4_ LLIJ 

Object  with  Object  with 
image  at  4  image  at  B 


11 


f  (f  +  m) 

- 


whence 


/  dmn 
Vm  —  n 


at 


140.  Measurement  of  Image.  —  At  a  distance  a 
least  100 /,  so  as  to 
rank  as  a  distant 
object,  set  off  at 
right  angles  two 
marks  \  a  distant 
from  the  center  line. 
The  distance  between  the  two  images  on  the  screen  will  be 
\  f.  This  distance  is  most  accurately  measured  by  a  sliding 
lens  (microscope)  focussed  on  the  aerial  image. 

Any  other  submultiple  of  a  can  be  similarly  used. 

If  a  is  not  large  enough  to  be  ranked  as  distant,  then  the 


distance  apart  of  the  images  is  ^  instead  of  ^ 
we  can  find  /  as  follows  : 

av  v2 


a  +  v 


v  — 


a  —  v 


From  this 


This  result  is  accurate  to  within  about  5  per  cent. 

141.   Comparison    with    Standard    Lens,    with    distant 
object. 


110  THICK-LENS    OPTICS 

_  focal  length  of  standard  X  length  of  image  by  lens 
J  length  of  image  by  standard  lens 

(See  Ex.  16,  §  47.) 

142.  Double  Focus,  with  Equal  Distances  to  Object  and 
Image. 

Find  F  and  Ff  for  parallel  rays.  Place  object  and  screen 
at  distances  somewhat  less  than  J  the  distance  between 
F  and  F',  and  move  by  small  equal  increments  until  the 
object  and  image  are  equally  distant  from  F  and  F'  and 

Mdal  planes 
Object I   \ ^_        Image 

Focus  Lens  Foeiis 

the  image  distinct.     These  distances  will  be  the  focal  length 
required.     (See  §  47,  Ex.  15,  in  connection  with  §  74.) 

143.  Double  Focus,  with  Unequal  Distances  to  Object 
and  Image. 

|<- X- *~, 

Serein  with,  F 

illuminated  scale. 
Ob/eet 

Determine  F,  F'  for  parallel  rays,  and  then  determine 
x  and  y  for  distinct  image.  Then 

/  =  focal  length  =  \/xy  [See  §  75 

If  the  principal  focus  is  within  the  combination,  interpose 
between  object  and  combination  any  good  short  focus  posi- 
tive lens  and  adjust  till  the  image  is  reflected  back  by  a  mir- 
ror to  coincidence  with  the  object  (or  slightly  to  one  side  by 
tilting  the  mirror).  Then  the  image  will  be  at  the  prin- 
cipal focus  of  the  combination,  since  the  emerging  rays  are 
parallel  and  reflected  back  parallel  by  the  mirror.  Remove 
the  combination  and  find  the  image  made  by  the  positive 
lens.  Determine  this  point  (on  the  mounting)  relative  to 


EXPERIMENTAL    OBSERVATIONS 


111 


the  combination  before  it  was  removed :  it  will  be  the  prin- 
cipal focus  (F)  of  the  combination.  Find  F'  in  the  same 
way.  Then  find  x,  y,  as  before,  and/=  Vx  y.  (Microscope 
ocular.) 

144.   By  Movement  of  Lens.  —  Same  method  as  in  §  131. 


Diagram  showing  the  extreme  rays  from  a  point;  in  the  two  positions. 
e  =  distance  between  nodals;  a  =  distance  between  the  two  posi- 
tions of  the  lens;  I  =  distance  between  the  screen  and  object 


a 

=  v  —  u 

u 

+  v  +  e  =  I 

I  —  a  —  e 

1  +  a  -  e 

2 

2 

i 

2 

2 

•  / 

l+a  -  e    ' 

I  —  a  —  e 

••-  / 

70       1            9 

I     +  d 

A  72    i  vcry  sr 

[§74 


-  a2          I2  +  a2 


[§67 

i  i/  i  c/ 

Notice  the  correction  induced  by  the  thickness  of  the 
lens.    *(Conf.  §  131.) 

FOCAL  RADIUS  OF  A  NEGATIVE  LENS 

145.   With  Sun.  —  If  lens  is  deep  and  not  too  small, 
focal  length  =  b.     (Conf.  §  47,  Ex.  14.) 


112 


THICK-LENS    OPTICS 


This  is  an  uncertain  method,  on  account  of  the  indis- 
tinctness of  the  bright  patches. 


rmcwes  of 
holes,  Hct  in. 
apart 


Card  with 
Wholes  a  in.  apart 


146.   With  Sun.    /  = 


Ad 


d  =  diameter 


of  the  lens  aperture,  D  =  diameter  of  the  circle  of  light 
cast  by  the  lens  when  in  the  path  of  the  sun's  rays,  A  = 
distance  of  the  screen  from  the  lens,  0.0094  =  2  tan  of  the 
apparent  diameter  of  the  sun. 

If  the  lens  is  deep  and  not  too  small,  we  can  write, 
neglecting  the  0.0094  A, 

Ad 
J  ~~~  D  -  d 

which  becomes  /  =  A,  if  D  =  2  d,  as  in  §  145. 
Proof. 

Extreme  ray  from  P,  thru  one  edge 
Hand 


D 


Extreme  ray  from  Q  thrw 
one  edge  o/ihelens 


C     Virtual  im< 
qfsuris  di 


By  similar  triangles 


EXPERIMENTAL    OBSERVATIONS  113 

147.  With  Stronger  Positive  Lens.  —  Combine  the  two 
lenses  and  find  focus  F.     Find  focal  length  of  positive 
lens,  F'.     Then  (§  91) 

F  F' 

f  =  focus  of  negative  lens  =  -=7 — — ^ 

r     —  r 

The  positive  lens  should  be  so  chosen  as  to  make  the 
difference  F'  —  F  as  large  as  possible. 

148.  With  Positive  Lens  and  Comparison  of  Images.  — 
Focus  with  a  positive  lens  and  measured  image.     Interpose 
negative  lens  and  measure  new  image.     Call  the  magnifica- 
tion over  the  positive  image,  M . 

Move  the  negative  lens  a  small  distance,  D,  nearer  the 
screen,  and  measure  the  image,  calling  its  magnification 
over  the  positive  image,  M' '. 

Then  focal  length  of  neg.  lens  =  f  —  ^,  _    „ 
Proof:       j  +  1  =  M,     j  +   1  =  Mr 
Therefore,  since  identically  -^ — 


then  f  =  M^lM 

d,  df  =  distances  from  negative  lens  to  screen. 

If  we  move  the  combination  instead  of  the  negative  lens, 
and  call  M',  M  the  actual  magnifications  of  the  images  for 
two  positions  of  the  object,  this  method  will  apply  to  a 
microscope  objective,  or  to  the  ocular  (positive  or  negative) 
by  inverting  it. 

149.  With  Positive  Lens  and  Comparison  of  Images 
(Lindsay  Johnson  method). 

Focus  with  positive  lens  and  measure  image. 

Interpose  negative  lens  and  measure  new  image,  adjusting 


114  THICK-LENS   OPTICS 

until  the  magnification  over  the  positive  image  is  2;   call 
the  distance  between  screen  and  negative  lens,  a. 

Move  the  negative  lens  until  the  magnification  over  the 
positive  image  is  3.  Call  the  distance  between  the  nega- 
tive lens  and  the  screen,  b. 

Then  F  =  focal  radius  of  the  negative  lens  =  b  —  a. 
Proof:    By  preceding  case 

b  —  a         b  —  a  ' 
F  =  M^M  =  JTTT2  =  6  ~  « 

TO   LOCATE   THE   NODAL   POINTS 

150.  (a)  Determine  the  focal  radius  (§§  134-144)  and 
lay  off  this  distance  from  the  focal  point,  marking  the 
result  on  the  mounting.  (Conf.  §  57.)  This  is  the  node  of 
emergence. 

(6)  Locate  the  point  (by  twisting  the  lens  around  a 
vertical  ayis:  on  the  optical  bench)  around  which  the  lens 
can  be  turned  on  a  vertical  axis  without  displacing  the 
image.  This  is  the  node  of  emergence. 

If  the  nodal  of  emergence  is  beyond  the  center  of  rotation, 
the  image  will  move  in  a  contrary  direction  to  that  of  the 
back  of  the  lens,  and  vice  versa. 

This  point  can  also  be  determined  by  reflecting  the  image 
back  through  the  lens  to  coincidence  with  the  object  by 
means  of  a  mirror  (or  slightly  to  one  side  by  tilting  the 
mirror).  When  a  slight  movement  of  rotation  produces  no 
movement  of  the  reflected  image  the  axis  of  rotation  is  at 
the  nodal,  and  moreover  the  focus  of  the  lens  is  the  distance 
of  the  axis  of  rotation  from  the  object  (image). 

Reverse  the  lens  and  repeat  the  operation,  to  find  the 
other  nodal:  both  cases. 


EXPERIMENTAL    OBSERVATIONS 


115 


MAGNIFYING   POWER:    TELESCOPE 

151.  Visual   Comparison  of  Images.  —  Distant  object. 

Mag.  power  =  —       N  =  number  of  clapboards   (divi- 
n  sions   on   a   scale)   seen  with 

one  eye  (naked)  which  are 
covered  by  n  clapboards  seen 
with  the  other  eye  through 
the  telescope 

152.  Visual  Comparison  of  Images.  —  Near  object. 
Focus  the  telescope  on  a  very  distant  object,  and  then 

fix  in  front  of  the  object  glass  a  thin  convex  lens  of  low 
power  (a  spectacle  glass  of  about  2  m.  focal  length). 

The  telescope  is  then  pointed  to  a  scale  at  such  a  dis- 
tance that  the  divisions  appear  well  defined,  focussing  by 
moving  the  scale,  not  the  telescope  tube  length. 

N    a 
Mag.  power  =  —  •  j- 

\N,  n  =  same  as  in  §  151 

a  =  dist.  from  scale  to  object  glass  of  telescope 
b  =  dist.  from  scale  to  eye  of  observer 

Proof.  —  With  the  lens  in  front  of  the  telescope,  we  have 
practically  a  large  microscope,  in  which  the  magnification  is 

&+/    F 


f 


-  (§  108) 


b  =  dist.  at  which  the  scale  is  seen 
/   =  focal  length  of  eyepiece 
F  =  focal  length  of  object  glass 
a  =  focal  length  of  intruded  lens 

But  the  magnification  of  the  telescope  is  (§  110) 


116  THICK-LENS    OPTICS 

6  +  /    F        \b  =  distance    of    dis- 


F 

hi 


/        b          tinct  landscape  vision 


a 


To  convert  the  first  into  the  second  we  must  multiply  by  7- 


u 
Q.E.D. 


MAGNIFYING   POWER    OF   A   MICROSCOPE 


153.  Visual  Comparison  of  Images.  —  With  one  eye 
(naked)  count  the  divisions  on  a  scale  10  in.  (25  cm.)  from 
the  eye  which  are  covered  by  one  or  more  divisions  on  a 
scale  seen  through  the  microscope. 


™ 

Mag.  power  =  — 


n 


N  =  number  of  divisions  seen  by 
the  naked  eye  at  10  in. 

n  =  number  of  divisions  seen  in 
the  same  space  through  the 
microscope 

A  convenient  way  for  observing  N  is  to  use  a  camera 
lucida  on  the  eyepiece,  with  the  naked-eye  scale  10  in. 
from  the  eye,  through  the  camera  lucida. 

154.  To  find  the  Work  done  by  the  Ocular.  —  Focus  on 
the  object  with  the  ocular  in  place.  Remove  the  ocular 
and  with  the  aid  of  a  small  lens  and  piece  of  ground  glass 
(see  §  156)  find  the  position  of  the  real  image  made  by  the 
objective.  Measure  its  distance  from  the  top  of  the  tube, 
and  note  the  corresponding  place  on  the  ocular.  Having 
previously  determined  the  focal  length  of  the  eye  and  field 
lenses  of  the  ocular,  calculate  (graphically  or  algebraically) 
the  reduction  caused  by  the  field  lens.  (The  position  of 
this  reduced  image  should  be  just  inside  the  focus  of  the 
eye  lens.  See  diagram  of  §  37.)  Multiply  this  by  the 
magnification  caused  by  the  eye  lens.  This  product  is 
the  final  action  of  the  ocular. 


EXPERIMENTAL    OBSERVATIONS  117 

If  the  first  focussing  is  done  on  a  scale,  on  or  in  the  posi- 
tion of  the  ground  glass,  it  will  give  the  magnification  caused 
by  the  objective. 

The  product  of  the  two  magnifications  should  equal  that 
found  by  §  153. 

(Second  method.)  Place  a  small  rectangular  opening  of 
known  width  at  one  end  of  a  tube  10  in.  long,  in  the  other 
end  of  which  is  placed  the  ocular  to  be  measured. 

At  the  image  of  the  rectangular  opening  (slightly  above 
the  ocular)  place  a  scale  and  with  a  lens  read  the  width  of 
the  image  on  the  scale.  The  reduction  from  rectangular 
opening  to  image,  inverted,  will  be  the  magnification  of 
the  ocular. 

,  ,  width  of  rectangular  opening 

Mag.  power  =  — r-rrr — j~r-  —r^    — y-5 

width  of  image  on  the  scale 

Proof.  —  The  details  are  left  to  the  reader,  with  the  fol- 
lowing guide.  Make  a  diagram  (skeleton,  §§  25,  77)  of  the 
ocular  and  by  §  61,  illustrated  by  §  95,  construct  the  nodal 
lines.  Starting  with  the  virtual  image  (call  it  A)  10 
inches  to  the  left  of  the  right-hand  lens  of  the  ocular,  derive 
from  it  the  aerial  object  (call  it  B),  just  outside  the  lens, 
which  produced  the  virtual  image.  The  ratio  of  magnifica- 
tion between  these  will  be  the  magnifying  (reduction  by 
first  lens,  and  final  magnification  by  second  lens)  power  of 
the  ocular.  For,  a  real  object,  the  rectangle,  at  A,  since 
these  are  conjugate  points,  will  give  a  real  image  at  B, 
which  can  be  measured  by  a  scale,  and  the  ratio  between 
this  A  and  B  will  be  the  same  as  before. 

Notice,  due  to  the  crossing  of  the  nodals,  the  aerial 
object  B  gives  a  virtual  image  at  A,  since  B  is  to  the  left 
of  the  H  nodal  line  and  within  the  F  focus,  though  outside 
the  lens.  (See  §  95,  Ex.  4.) 


118  THICK-LENS    OPTICS 

When  the  objective  is  in  operation  the  aerial  object  is 
generally  within  the  ocular  combination,  instead  of  outside, 
and  the  virtual  image  at  infinity.  But  this  does  not 
materially  alter  the  angle  under  which  the  virtual  image  is 
seen,  and  therefore  not  the  magnifying  power. 

Check  this  by  locating  A  and  B  through  the  lenses, 
instead  of  through  the  nodal  planes.  Pass  A  toward  the 
left  to  infinity,  and  note  how  B  passes  through  the  lens 
to  a  point  between  the  lenses;  the  point  to  which  the 
image  made  by  the  objective  is  deflected  by  the  first  lens 
of  the  ocular.  The  aerial  image  at  this  point  gives  the 
virtual  image  at  infinity  seen  by  the  eye  of  the  observer. 

See  also  method  of  §  143. 

155.  To  Find  the  Index  of  Refraction.  —  Sight  the  ob- 
jective of  a  microscope  on  a  well-marked,  hard  (celluloid) 
white  surface,  or  piece  of  scratched  glass,  and  read  the 
scale  on  the  limb  of  the  microscope  (vernier  attachment 
necessary),  the  aperture  of  the  objective  having  been 
diminished  by  slipping  over  the  objective  a  cap  (paper) 
with  a  small  central  opening  (1  mm.  =  ^V  in-  diameter). 

Having  marked  the  center  of  the  lens  whose  index  of 
refraction  is  desired,  by  a  small  circular  ink  spot  with  a 
clear  center  on  the  upper  surface  (in  order  to  locate  the 
center  through  the  microscope),  slip  the  lens  under  the 
objective  and  focus  on  the  center  of  the  upper  surface. 
The  difference  of  the  readings  is  the  thickness  (t)  of  the  lens. 

Then  carefully  focus,  through  the  lens,  on  the  marked 
white  surface  on  which  the  lens  rests.  The  difference  of 
the  last  two  readings  will  be  a,  t  —  a  being  the  amount 
the  objective  must  be  raised  from  focussing  on  the  white 
surface,  due  to  the  interposition  of  the  lens.  Then 

-  1    r  -  a 
a    r  —  t 


EXPERIMENTAL    OBSERVATIONS  119 

or  if  the  lens  is  merely  plane  glass 

_  t_ 

=  index  of  refraction 

t  =  thickness  of  the  lens 

a  =  amount  the  objective  must  be  lowered  from  focus- 
sing on  the  top  of  the  lens,  to  bring  the  mark  on 
the  white  surface  into  focus 

r  =  radius  of  curvature  of  the  upper  surface  of  the  lens 

N.B.  —  To  get  satisfactory  results,  very  great  care  and 
a  fine  vernier  are  necessary. 
Proof. 


Diagram  showing  the  lens  in  place  under  the  ob- 
jective. 0  =  center  of  curvature  of  the  upper  surface. 
S  =  marked  spot  on  the  white  surface  on  which  the 
lens  rests,  a  =  distance  the  stage  must  be  raised  or 
the  objective  lowered  in  focussing  from  the  top  of  the 
lens  to  the  marked  spot  S  (through  the  lens),  a  = 
angle  of  incidence,  /8,  of  refraction,  d  =  semi-opening 
of  lens,  c  =  semi-diameter  of  visible  part  of  lens. 

Note.  —  a,  /?,  and  y  are  supposed  to  be  so  small  (due  to 
the  narrowed  opening  of  the  objective)  that  their  cosines 
=  1  and  their  sines  =  a,  ft,  y,  respectively,  as  also  their 
tangents  =  a,  /?,  y.  (See  any  trigonometry,  Functions  of 
Small  Angles.) 


120 


THICK-LENS   OPTICS 


sin  (y  +  a)    _  t_ 
sin  (y  +  ft)  ~  a 

sin  y  cos  a  +  cos  y  sin  a  _  ^  _ 
sm  y  cos  ft  +  cos  y  sin  ft      a      c 


[§6 


-  cos  a  4-  sin  a 


-  cos  ft  +  sin 
r 


[sin  y  = 


sin  a 


sm  a 
sin  ft 


c 

-  -\-srn  ft 
r 

,  I  + sin  ^ 

i  T 

a     sin  ft 


[ 


COS  a 

cos 


tc 


a  r  sin  /: 

+  --        ° 
a      r  sin 


ra 


L  raa        ^ 

a  raa  —  tc  +  ta  ) 

t  r  —  a 

a  r  —  t 


ra  sn 


raa 
c 


fna  =  a, 
ex 
/D 
AJ     ^"^    — 
tj^ 


—  a,  etc. 


tan  (y  +  a) 

_d 

~  y  i        ,,. 

c 

tan  y 

r 

d 

d       c       dr  —  cu 

a 

u 

*  ~  u      r           ur 

C 

_d 

a 

u 

EXPERIMENTAL    OBSERVATIONS  121 

If  r  =  oo,    /a  =  - 

Check  your  determination  of  /*  by  using  it  to  calculate 
F  of  the  lens,  and  test  by  observation. 

Similarly  for  a  concave  surface,  we  would  have  [since 
the  angle  (y  +  a)  becomes  replaced  by  (a  —  y)  and 


PRACTICAL   SUGGESTIONS 

156.  To  Focus  Accurately.  —  Set  a  fine  pin  (needle)  in 
line  with  the  lens,  and  with  the  eye  in  line  fix  the  vision 
on  the  pin.     Adjust  the  lens  backward  or  forward  until 
a  motion  of  the  head  slightly  sidewise  does  not  alter  the 
position  of  the  pin  on  the  image  seen  through  the  lens. 
If  on  moving  the  head  the  pin  moves  across  the  image  in 
the  same  direction,  the  image  (and  therefore  the  lens)  is 
too  close,  and  vice  versa. 

Better  still,  use  a  short  focus  lens,  focussing  it  on  the 
edge  of  a  translucent  (transparent)  screen  (piece  of  cellu- 
loid). Then  move  the  lens  until  the  image  appears  dis- 
tinct, testing  similarly  by  the  motion  jy 
of  the  head. 

157.  To  Measure  an  Angle  with- 
out Angular  Instruments.  —  Deter- 
mine three  points,  A,  at  the  vertex £T~ 

of  the  angle,  B,  in  line  with  one  side  of  the  angle,  and  C, 
in  line  with  the  other  side  and  at  right  angles  to  AB. 

This  may  be  done  on  the  ground,  or  on  a  table  or  top 
of  a  level  box.     Then 


90" 


122  THICK-LENS    OPTICS 

BC  BC 

tan  A  =35     or    sin  A  -=  -^ 

whence  the  angle  A  can  be  found  by  a  table  of  tangents  or 
sines. 

158.  Make  the  experimental  observations  with  mono- 
chromatic light  (as  well  as  white)  by  using  red,  blue,  green 
screens  (colored  glass). 

Use  no  alcohol  or  other  solvent  on  mounted  lenses, 
except  in  an  emergency  and  with  the  greatest  care.  Cleanse 
greasy  lenses  with  a  weak  solution  of  washing  soda,  rinsing 
with  clean  water. 

Keep  all  lenses  covered  from  dust;  keep  a  cover  over 
the  eyepiece  of  the  microscope  when  not  in  use. 

Keep  lenses  out  of  the  sun  as  much  as  possible  —  they 
will  gradually  discolor.  The  cement  may  overheat. 

Clean  lenses  with  the  greatest  care,  lightest  pressure, 
softest  cloth  free  from  dust  and  grit,  with  a  circular  motion, 
never  across  the  lens.  Use  soft  camel's  hair  brush  when 
feasible. 

159.  Home-Made   Optical  Bench.  —  A  little  skill  can 
make  a  home-made  optical  bench,  as  shown,  with  which 
fairly  good  work  can  be  done.     In  the  absence  of  anything 
better,  lenses  fastened  on  the  tops  of  corks  with  pins  will 
do  roughly  good  work. 

a,  board  on  edge  (J  or  §  in.  wide),  or  graduated  yard 
stick. 

6,  pedestal  to  keep  board  upright,  two  or  more. 

c,  sliding  piece  on  top  of  board,  with  center  line  for  read- 
ing distances  moved. 

d,  center  line  of  the  various  pieces. 

e,  rotating    piece,    pivoted    to    c    at    center,    by    small 
screw    or    pivot,    exactly    in    center,    and    in    line    with 
center  line   d. 


EXPERIMENTAL    OBSERVATIONS 


123 


/,  lens  carrier  sliding  on  e,  so  as  to  allow  bringing  the 
nodals  over  the  center  of  revolution. 

g,  g,  sloping  sides  to  accommodate  different  sized  lenses. 


h,  h 


grooves  in  which  to  slip  the  lens  so  that  it  will  be 
held  upright:  h  is  used  for  a  short  focus  lens,  when 


i  would  be  too  far  from  the  end  of  /,  the  nearest 
approach  of  the  screen. 
j,  center  line  of  h  h,   a  known  distance  from  d. 
Suggestions  as  to  how  to  put  the  pieces  together  are 
indicated  by  dotted  lines. 

Modified  carriers  (without  rotation)  should  be  provided 
for  holding  screens,  reading  lenses  (§  156),  etc.,  to  be  used 
in  connection  with  the  one  above. 


124 


THICK-LENS    OPTICS 


-5 


Kg] 3 


Fig  14    C 


Fig.  15 


-r 


APPENDIX 

THIS  contains  a  series  of  progressive  propositions  giving 
succinct  methods  of  construction  and  interpretation,  with 
important  theorems  for  surfaces  and  lenses,  culminating  in 
propositions  XIII-XVIII,  which  give  general  discrimina- 
tions for  locating  the  nodals  and  foci  for  different  forms  of 
lenses. 

These  propositions  afford  a  valuable  general  check  upon 
the  calculations  and  graphical  constructions,  guided  as  they 
are  by  the  numerical  relations  between  the  surfaces. 

The  calculator  cannot  have  too  many  checks,  as  he  will 
quickly  discover  when  he  essays  an  independent  and  uncor- 
roborated investigation. 

These  propositions  give  a  complementary  point  of  view 
to  that  in  the  body  of  the  text,  valuable  and  almost  in- 
dispensable, especially  for  those  making  original  numerical 
investigations. 

SURFACE    REFRACTION 

Notation.  —  Prolong  is  the  prolongation  on  the  right  of 
the  surface  (light  is  always  supposed  to  come  from  the  left) 
of  the  ray  impinging  on  the  left  side. 

Emerge  is  the  position  of  the  ray  after  refraction  through 
the  second  surface  from  the  denser  to  the  rarer  medium. 

r  =  the  radius  of  the  surface  of  incidence,  the  surface  to 
the  left  of  the  denser  medium. 

s  =  the  radius  of  the  surface  of  emergence,  the  surface  to 
the  right  of  the  denser  medium. 

e  =  thickness  of  the  lens. 

Hf,  Ff  =  nodal  of  emergence  and  corresponding  focus. 

H,  F  =  nodal  of  incidence  and  corresponding  focus. 

[For  convenience  of  illustration,  /*  is  assumed  =  f .     If  /* 


126  LENS   OPTICS 

is  not  equal  to  f ,  in  the  results  which  follow  substitute 
^     for  3,  and  -  — r  for  2.] 

I.  To  trace   an  incident  ray,   surface  refraction.     Figs. 
1  and  2.     (Conf.  §33.)     Note  carefully  the  order  of  the 
letters,  which  indicate  the  order  of  construction;  and  the 
formula  for  construction,  rs  — ->  <fo  1 1  • 

II.  To  trace  an  emergent  ray,  surface  refraction.     Figs. 
3  and  4. 

III.  (Cor.  to  II.)     Fig.  5.     Emergent  rays  from  a  surface 
with  a  +  radius,  with  the  prolongs  convergent  to  a  point  on 
the  axis 

beyond  the  3s  point,  are  bent  upward,  above  the  hori- 
zontal ; 

at  the  3  s  point,  emerge  horizontal; 
within  the  3  s  point 

and  beyond  the  s  point,  are  bent  upward  above  the  pro- 
long, emerging  convergent: 

and  within  the  s  point,  are  bent  downward,  below  the 
prolong,  emerging  convergent. 

IV.  (Cor.  to  III.)     Emerges  originating  from  points  on 
the  axis  to  the  left  of  a  +  radius  surface  are  divergent,  bent 
upward  from  the  prolong. 

V.  (Cor.  to  III.)     Fig.  6.     H3s>3r-e  (positive  me- 
niscus, convex  to  the  rays),  then  incident  horizontal  rays 
emerge  convergent,  but 

rising  from  the  prolong,  if  s  <  3r  —  e', 

falling  from  the  prolong,  if  s  >  3  r  —  e. 

If  3s  <  3r  —  e  (negative  meniscus,  convex  to  the  rays), 
then  the  incident  horizontal  rays  emerge  divergent,  rising 
from  the  prolong. 

VI.  (Cor.  to  II.)     Fig.  7.     Emerges  from  a  —  radius  sur- 
face, originating  from  a  point  on  the  axis  to  the  left  of  the  3  s 
point,  are  bent  downward  to  convergence: 

from  the  3s  point,  are  bent  downward  to  horizontally; 

from  a  point  within  the  3  s  point,  are  divergent,  falling  be- 
low the  prolong,  if  from  without  the  s  point;  rising  above 
the  prolong,  if  from  within  the  s  point. 

VII.  (Cor.  to  VI.)     Rays  convergent  to  a  point  on  the 


APPENDIX  127 

right  of  a  —  radius  surface  have  their  emerges  bent  down 
below  the  prolongs. 

VIII.  (Cor.  to  VI  and  I.)     Fig.  8.     If8s-e>8r  (neg- 
ative meniscus,  concave  to  rays),  then  incident  horizontal 
rays  are  divergent; 

falling  from  the  prolong,  if  s  <  3  r  +  e; 

rising  from  the  prolong,  if  s  >  3  r  +  e. 

li  3s  —  e  <  3r  (positive  meniscus,  concave  to  the  rays), 
then  incident  horizontal  rays  are  convergent,  falling  below 
the  prolong. 

IX.  (Cor.  to  VIII.)     Fig.  9.     In  a  positive  meniscus,  con- 
cave to  the  rays,  (s  <  r  +  e/8),  the  nodal  of  emergence  is 
outside  the  lens  to  the  right,  and  F'  to  the  right  of  that. 

X.  (Cor.  to  VIII.)     Figs.  10,  11.     In  a  negative  menis- 
cus, concave  to  the  rays  (s  >  r  +  e/3),  the  nodal  of  emer- 
gence is  on  the  outside  of  the  lens  to  the  left  if  s  <3 r  +  e\  or 
in  the  lens  if  s  >  3  r  +  e.     In  either  case  F'  is  to  the  left 
of  H'. 

XI.  (Cor.  to  V.)     In  a  positive  meniscus,  convex  to  the 
rays,  since  3  s  >  3  r  —  e,  the  nodal  of  emergence  is  outside  the 
lens  to  the  left  if  s  <  3  r  —  e;  inside  the  lens  if  s  >  3  r  —  e. 

In  either  case  the  focus  F'  is  to  the  right. 

XII.  (Cor.  to  V.)     In  a  negative  meniscus  (3  s  <  3  r  —  e), 
convex  to  the  rays,  since  the  emerge  rises  from  the  prolong, 
the  nodal  of  emergence  is  to  the  right  and  outside,  and  F' 
to  the  left. 

XIII.  (Cor.    to    I     and    IV.)     Double    concave    lens. 
Emerges  resulting  from  incident  horizontal  rays  are  bent 
upward  from  the  prolong  and  H'  is  within  the  lens,  Ff  to  the 
left.     Similarly  as  to  H,  F,  mutatis  mutandis. 

XIV.  (Cor.  to  I  and  VII.)     Double  convex  lens.     Inci- 
dent horizontal  rays  are  bent  down  by  the  first  surface  and 
the  emerge  falls  below  the  prolong,  therefore  H'  is  within  the 
lens,  and  F'  to  the  right.     Similarly  as  to  H,  F. 

XV.  (Cor.  to  IX  and  XL)     Positive  meniscus,  concave 
to  the  rays.     Hf  is  outside  the  convex  with  F'  further  out- 
side, measured  with  the  rays. 

H  is  outside  the  convex  if  r  <  3  s  —  e. 
H  is  inside  if  r  >  3  s  —  e. 


128  LENS   OPTICS 

F  is  measured  from  H  toward  the  concave,  against  the 
rays. 

XVI.  (Cor.  to  X  and  XII.)     Negative  meniscus,   con- 
cave to  rays. 

H'  is  outside  the  concave  if  s  <  3  r  +  e. 

Hr  is  inside  if  s  >  3  r  -\-  e. 

F'  is  measured  from  H'  against  the  ray. 

H  is  outside  the  concave  with  F  measured  with  the  rays. 

XVII.  (Cor.  to  X  and  IX.)     Positive  meniscus,  concave 
toward  the  rays. 

H'  is  outside  the  convex  if  s  <  3  r  —  e. 

Hf  is  inside  if  s  >  3  r  —  e. 

F'  is  measured  with  the  rays. 

H  is  outside  the  convex  with  F  measured  against  the  rays. 

XVIII.  (Cor.  to  XII  and  X.)     Negative  meniscus,  con- 
vex to  rays. 

H'  is  outside  the  concave,  with  F'  measured  against  the 
rays. 

H  is  outside  the  concave  if  r  <  3  s  +  e. 
H  is  inside  if  r  >  3  s  +  e. 
F  is  measured  with  the  rays. 

SURFACE    REFLECTION 

(The  order  of  the  letters  in  the  diagrams  indicates  the  order  of  oper- 
ations. Conf.  :§  7.) 

Reflection  from  convex  surface.  Figs.  12,  13,  14.  Image 
virtual  and  smaller.  (Conf.  notation  of  prop.  I.) 

Reflection  from  concave  surface.  Figs.  15,  16.  Image 
real. 

Reflection  from  concave  surface.  Fig.  16.  Object  out- 
side the  center.  Image  real,  inverted,  smaller,  inside  the 
object. 

Reflection  from  concave  surface.  Fig.  17.  Object  in- 
side the  focus  (r/2).  Image  virtual,  erect,  larger,  behind  the 
mirror.  (For  object  between  the  center  and  focus,  image 
is  real,  inverted,  larger,  outside  the  object.) 


INDEX 


A,  A',  75 

AN,  A'N',  75 

AN  +  W,  A'N'  +  91',  75 

Analytical  investigation,  55 

Axis  of  Surface,  3 


B 


1,67 


Back     focal     distance,     two     thick 

lenses,  71 

two  thin  lenses,  67 
light  from  right,  70 


Camera  magnification,  89 
Center,  optical,  9,  43 
Circle  of  confusion,  35 
Combinations  of  lenses,  64 
Condenser,  best  position,  32 
Confusion,  circle  of,  35 
Copying,  31 

D 
D,  25 

Diagrammatic  investigations,  9 
Diagrammatic  procedure,  10 
Diopters,  25 


e,  41,  42,  125 
e,  67,  75 
Emerge,  125 

Emergence,  principal  point  of,  39 
Enlarging,  31 
Equation,  of  line,  56 
refracted  ray,  57 


Equivalent  focus,   two  thin  lenses, 

67 
Equivalent  focal  length,  two  thick 

lenses,   71 
Equivalent      thickness,      thick-lens 

combination,  72 
Equivalent  thin-split,  42,  74 
Exposure,  33 
Eye,  48 


F,  Fr,  7,  75,  125 
/,  5,  34,  75 
/',  7,  75 
/  number,  34 

/i»  /z»  focal  length  of  lenses  of  com- 
bination, 67,  75 
Focal     length,    radius,    two     thick 

lenses,  71 

telephoto  combination,  91 
thin  positive  lens,  by  sun,  104 
by  lens  distances,  105 
with  a  telescope,  105 
by  different  positions  of  lenses, 

105 
by  equality  of  object  and  image, 

105 

by  comparison  of  images,  105 
thick    positive     lens,    by     highly 

magnified   image,  107 
by   swing   of   camera,    107 
by  movement  of  screen,  108 
by  angle  of  vision,   108 
by  unit  screen  movement,  108 
by  measurement  of  image,  109 
by    comparison    with    standard 

lens,  109 
by  double  focus,  110 


130 


INDEX 


by  movement  of  lens,  111 
of  negative  lens,   by  sun,    111 
by  stronger  positive  lens,  113 
by  positive  lens  and  compari- 
son of  images,   113 
Focal  points,  62 
Focus,  back  focal,  two  thin  lenses, 

67 

equivalent,  two  thin  lenses,  67 
to  accurately,   121 
Formulae,  use  of,  54 

use  of  for  combined  lenses,  74 


Graphic    check   on    calculation,    22 
Graphic  construction,  76 

tracing  any  ray,  54 

oblique  ray,  15 

H 

H,  H',  41,  42,  48,  60,  125 
Human  eye,  48 
Hyperfocal  distance,  34 


Index  of  refraction,  1,  118* 

L 
Line,  Equation  of,  56 

M 

M,  29 

H,  index  of  refraction,  2 
Magnification,  convex  lens,  28 
camera,  28,  89 
compound  microscope,  87 

by  visual  comparison  of  images, 

116 

simple  microscope,  37 
ocular,  111,  116 
opera  glass,  89 
surface  reflection,  100 
telescope,  88 

by  visual  comparison  of  images, 
115 


Microscope,    simple,    magnification, 

37 
compound,  magnification,  87 

N 

N,  31 
W,  W,  75 
Nodal,  49 

Nodal  distances,  thick  lenses,  72 
Nodal  plane,  images  in,  47 
Nodal  points,  42,  61 

construction  for,  42,  44 

calculation  for,  45 

to  locate,  114 
Number  of  the  lens,  25 


Oblique  rays,  diagram  for,  15,  54 
Ocular,  magnification,  111,  116 
Opera  glass,  magnification,  89 
Optical  bench,  home  made,  122 
Optical  center,  9,  43 


P,  P',  25 

Point  of  emergence,  39 

Powers  of  distances,  25 

lenses,  25 

not  in  contact,  69 
thick-lens  combination,  73 
Principal  points,  39,  60 

point  of  emergence,  39 

plane,  41,  60 
Prolong,  125 

Q 

Q,  Q',  25 

R 

r,  5,  7,  125 
Radius  of  curvature  of  surface,  102 

by  matching,  103 

small  curvature,  104 
Reduction  factor,  negative  lens,  36 
Reduction  factor,  telephoto,  97 
Reflection  at  spherical  surfaces,  98 

diagrammatic  construction,  128 
Refracted  ray,  equation  of,  57 
Refraction,  index  of,  1,  118 

surface,  1,  2,  126 


INDEX 


131 


s,  6,  7,  125 
Sine,  1 

Sines,  Law  of,  2 
Slowness  factor,  34 
Spectacles,  26,  27 
Standard  formula,  18 
Surface  refraction,  1,  2,  125 
Equation  for,  4 


Tangent,  1 

Telephoto  combination,  focal  length, 

91 

focal  distance,  92 
focal  radius,  94 
distance  for  given  magnification, 

96 

lens,  90 
magnification,  91 


reduction  factor,  97 
Telescope,  magnification,  88 
Thick  lens,  39 
Thin  lenses,  in  contact,  64 

not  in  contact,  65 
Thin-split,  42,  74 

U 


U.S.  number,  33,  34 

Use  of  formula  ---  =  -   18 
v      u      f 


Vertex  of  Surface,  3 

W 

w,  5 


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This  list  includes  the  technical  publications  of  the  following  English 
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Abbott,  A.  V.     The  Electrical  Transmission  of  Energy 8vo,  *$5  oo 

A  Treatise  on  Fuel.     (Science  Series  No.  9.) i6mo,  o  50 

Testing  Machines.     (Science  Series  No.  74.) i6mo,  o  50 

Adams,  H.     Theory  and  Practice  in  Designing .8vo,  *2  50 

Adams,  H.  C.    Sewage  of  Seacoast  Towns 8vo,  *2  oo 

Adams,  J.  W.     Sewers  and  Drams  for  Populous  Districts — 8vo,  2  50 

Adam,  P.     Practical  Bookbinding.     Trans,  by  T.  E.  Maw.  i2mo,  *2  50 

Addyman,  F.  T.     Practical  X-Ray  Work .  8vo,  *4  oo 

Adler,  A.  A.    Principles  of  Projecting  Line  Drawing 8vo, 

Vol.  I.  Theory  of  Engineering  Drawing 

Aikman,  C.  M.     Manures  and  the  Principles  of  Manuring.  .  .  8vo,  2  50 

Aitken,  W.     Manual  of  the  Telephone 8vo,  *8  oo 

d'Albe,  E.  E.  F.     Contemporary  Chemistry i2mo,  *i  25 

Alexander,  J.  H.     Elementary  Electrical  Engineering i2mo,  2  oo 

Universal  Dictionary  of  Weights  and  Measures 8vo,  3  50 

"  Alfrec."     Wireless  Telegraph  Designs 

Allan,    W.     Strength    of    Beams    under     Transverse    Loads. 

(Science  Series  No.  19  ) i6mo,  o  50 

— ! Theory  of  Arches.     (Science  Series  No.  u.) i6mo, 


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Allen,  H.     Modern  Power  Gas  Producer  Practice  and  Applica- 
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Anderson,  F.  A.     Boiler  Feed  Water 8vo,  *2  50 

Anderson,  Cap.  G.  L.     Handbook  for  the  Use  of  Electricians  .8vo,  3  oo 

Anderson,  J.  W.     Prospector's  Handbook i2mo,  i  50 

Andes,  L.     Vegetable  Fats  and  Oils 8vo,  *4  oo 

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Annual  Reports  on  the  Progress  of  Chemistry. 

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Argand,  M.     Imaginary  Quantities.     Translated  from  the  French 

by  A.  S.  Hardy.     (Science  Series  No.  52.) i6mo,  o  50 

Armstrong,  R.,  and  Idell,  F.  E.     Chimneys  for  Furnaces  and 

Steam  Boilers.     (Science  Series  No.  i.) i6mo,  o  50 

Arnold,  E.     Armature  Windings  of  Direct  Current  Dynamos. 

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Ashe,  S.  W.,  and  Keiley,  J.  D.     Electric  Railways.     Theoreti- 
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Atkinson,  A.  A.     Electrical  and  Magnetic  Calculations 8vo,  *i  50 

Atkinson,   J.   J.     Friction  of  Air   in   Mines.     (Science   Series 

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Atkinson,  J.  J.,  and  Williams,  E.  H.,  Jr.     Gases  Met  with  in 

Coal  Mines.     (Science  Series  No.  13.) i6mo,  o  50 

Atkinson,  P.     The  Elements  of  Electric  Lighting i2mo,  i  50 

The  Elements  of  Dynamic  Electricity  and  Magnetism.  i2mo,  2  oo 

Power  Transmitted  by  Electricity i2mo,  2  oo 

Auchincloss,  W.  S.     Lime  and  Valve  Motions  Simplified 8vo,  *i  50 

Ayrton,  H.     The  Electric  Arc 8vo,  *s  oo 

Bacon,  F.  W.     Treatise  on  the  Richards  Steam-Engine  Indica- 
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Bailes,  G.  M.  Modern  Mining  Practice.  Five  Volumes. 8vo,  each,  3  50 

Bailey,  R.  D.     The  Brewers'  Analyst 8 vo,  *s  oo 

Baker,  A.  L.     Quaternions i2mo,  *i  25 

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Baker,  Benj.     Pressure  of  Earthwork.     (Science  Series  No.  56.) 

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Baker,  J.  B.     Magneto  and  Sparking  Coil (In  Press.) 

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Sewerage  and  Sewage  Purification.     (Science  Series  No.  18.) 

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Ball,  R.  S.     Popular  Guide  to  the  Heavens 8vo,  *4  50 

Natural  Sources  of  Power.     (Westminster  Series) 8vo,  *2  oo 

Ball,  W.  V.     Law  Affecting  Engineers 8 vo,  *3  50 

Bankson,  Lloyd.    Slide  Valve  Diagrams.     (Science  Series  No. 

108.) i6mo,  o  5d 

Barba,  J.     Use  of  Steel  for  Constructive  Purposes i2mo,  i  oo 

Barham,  G.  B.    Development  of  the  Incandescent  Electric 

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Barker,  A.     Textiles  and    Their    Manufacture.     (Westminster 

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Barker,  A.  H.     Grapic  Methods  of  Engine  Design 12010,  *i  50 

Barnard,  F.  A.  P.     Report  on  Machinery  and  Processes  of  the 
Industrial  Arts  and  Apparatus  of  the  Exact  Sciences  at 

the  Paris  Universal  Exposition,   1867 8vo,  5  oo 

Barnard,  J.  H.     The  Naval  Miliatiaman's  Guide.  .i6mo,  leather,  125 
Barnard,  Major  J.  G.     Rotary  Motion.     (Science  Series  No.  90.) 

i6mo,  o  50 

Barrus,  G.  H.     Boiler  Tests 8vo,  *3  oo 

Engine  Tests 8vo,  *4  oo 

The  above  two  purchased  together *6  oo 

Barwise,  S.     The  Purification  of  Sewage i2mo,  3  50 

Baterden,  J.  R.     Timber.     (Westmenster  Series) 8vo,  *2  oo 

Bates,  E.  L.,  and  Charlesworth,  F.     Practical  Mathematics  and 

Geometry  for  Technical  Students I2mo, 

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Beadle,  C.  Chapters  on  Papermaking.  Five  Volumes. i2mo,  each,  *2  oo 

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Bechhold.     Colloids  in  Biology  and  Medicine.     Trans,  by  J.  G. 

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Bedell,  F.,  and  Pierce,  C.  A.  Direct  and  Alternating  Current 

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Beech,  F.  Dyeing  of  Cotton  Fabrics 8vo,  *3  oo 

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Beckwith,  A.     Pottery 8 vo,  paper,  o  60 

Beggs,  G.  E.    Stresses  in  Railway  Girders  and  Bridges ....  (In  Press.) 

Begtrup,  J.     The  Slide  Valve 8vo,  *2  oo 

Bender,  C.  E.     Continuous  Bridges.     (Science  Series  No.  26.) 

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Bennett,  H.G.      The  Manufacture  of  Leather 8vo,  *4  50 

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Bernthsen,  A.     A  Text-book  of  Organic  Chemistry.     Trans,  by 

G.  M'Gowan i2mo,  *2  50 

Berry,  W.  J.     Differential  Equations  of  the  First  Species. 

i2mo  (In  Preparation.) 


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Bersch,  J.     Manufacture  of  Mineral  and  Lake  Pigments.     Trans. 

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Bertin,  L.  E.     Marine  Boilers.     Trans,  by  L.  S.  Robertson.  .8 vo,  500 

Beveridge,  J.     Papermaker's  Pocket  Book i2mo,  *4  oo 

Binns,  C.  F.      Ceramic  Technology 8vo,  *5  oo 

Manual  of  Practical  Potting 8vo,  *7  50 

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Birchmore,  W.  H.    Interpretation  of  Gas  Analysis i2mo,  *i  25 

Elaine,  R.  G.     The  Calculus  and  Its  Applications i2mo,  *i  50 

Blake,  W.  H.     Brewers'  Vade  Mecum 8vo,  *4  oo 

Blake,  W.  P.     Report  upon  the  Precious  Metals 8vo,  2  oo 

Bligh,  W.  G.     The  Practical  Design  of  Irrigation  Works 8vo,  *6  oo 

Blucher,  H.     Modern  Industrial  Chemistry.    Trans,  by  J.  P. 

Millington 8vo,  *7  50 

Blyth,  A.  W.     Foods:  Their  Composition  and  Analysis 8vo,  7  50 

Poisons :  Their  Effects  and  Detection 8 vo,  7  50 

Bb'ckmann,  F.     Celluloid i2mo,  *2  50 

Bodmer,  G.  R.     Hydraulic  Motors  and  Turbines I2mo,  5  oo 

Boileau,  J.  T.     Traverse  Tables , 8vo,  5  oo 

Bonney,  G.  E.     The  Electro- platers'  Handbook i2mo,  i  20 

Booth,  W.  H.     Water  Softening  and  Treatment 8vo,  *2  50 

Superheaters  and  Superheating  and  their  Control.  .  .  .8vo,  *i  50 

Bottcher,  A.     Cranes:  Their  Construction,  Mechanical  Equip- 
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Bottler,  M.  Modern  Bleaching  Agents.  Trans,  by  C.  Salter.i2mo,  *2  50 

Bottone,  S.  R.      Magnetos  for  Automobilists i2mo,  *i  oo 

Boulton,  S.  B.     Preservation  of  Timber.     (Science  Series  No. 

82.) i6mo,  o  50 

Bourgougnon,  A.    Physical  Problems.     (Science  Series  No.  113.)  • 

i6mo,  o  50 
Bourry,  E.      Treatise   on  Ceramic    Industries.       Trans,   by 

W.  P.  Rix. 8vo,  *5  oo 

Bow,  R.  H.     A  Treatise  on  Bracing 8vo,  i  50 

Bowie,  A.  J.,  Jr.     A  Practical  Treatise  on  Hydraulic  Mining .  8vo,  5  oo 

Bowker,  W.  R.     Dynamo  Motor  and  Switchboard  Circuits ..  8vo,  *2  50 

Bowles,  0.     Tables  of  Common  Rocks.     (Science  Series)  .i6mo,  050 

Bowser,  E.  A.    Elementary  Treatise  on  Analytic  Geometry.  i2mo,  i  75 

Elementary    Treatise    on    the    Differential    and    Integral 

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Bowser,  E.  A.     Elementary  Treatise  on  Analytic  Mechanics 

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Elementary  Treatise  on  Hydro-mechanics I2mo,       2  50 

A  Treatise  on  Roofs  and  Bridges I2mo,     *2  25 

Boycott,  G.  W.  M.     Compressed  Air  Work  and  Diving 8vo,     *4  oo 

Bragg,  E.  M.     Marine  Engine  Design i2mo,     *2  oo 

Brainard,  F.  R.    The  Sextant.     (Science  Series  No.  ioi.).i6mo, 

Brassey's  Naval  Annual  for  1911 8vo,     *6  oo 

Brew,  W.     Three-Phase  Transmission 8vo,     *2  oo 

Brewer,  R.  W.  A.     Motor  Car  Construction i2mo,     *2  oo 

Briggs,  R.,  and  Wolff,  A.  R.  '  Steam-Heating.     (Science  Series 

No.  67.) i6mo,      o  50 

Bright,  C.     The  Life  Story  of  Sir  Charles  Tilson  Bright 8vo,     *4  50 

Brislee,  T.  J.    Introduction  to  the  Study  of  Fuel.    (Outlines  of 

Industrial  Chemistry.) 8vo,     *3  OO 

British  Standard  Sections 8x15     *i  oo 

Complete  list  of  this  series  (45  parts)  sent  on  application. 
Broadfoot,  S.  K.    Motors  Secondary  Batteries.     (Installation 

Manuals  Series.) i2mo,     *o  75 

Broughton,  H.  H.    Electric  Cranes  and  Hoists *g  oo 

Brown,  G.     Healthy  Foundations.     (Science  Series  No.  80.) 

i6mo,       o  50 

Brown,  H.     Irrigation 8vo,     *5  oo 

Brown,  Wm.  N.     The  Art  of  Enamelling  on  Metal i2mo,     * 

Handbook  on  Japanning  and  Enamelling i2mo,     * 


House  Decorating  and  Painting i2mo, 

History  of  Decorative  Art i2mo,     * 

Dipping,    Burnishing,    Lacquering    and    Bronzing    Brass 


oo 
50 
50 

25 


Ware i2mo,  *  oo 

Workshop  Wrinkles 8vo,  *  oo 

Browne,  R.  E.  Water  Meters.  (Science  Series  No.  81.).  i6mo,  o  50 

Bruce,  E.  M.  Pure  Food  Tests I2mo,  *i  25 

Bruhns,  Dr.  New  Manual  of  Logarithms 8vo,  half  mor.,  2  oo 

Brunner,  R.  Manufacture  of  Lubricants,  Shoe  Polishes  and 

Leather  Dressings.  Trans,  by  C.  Salter 8vo,  *3  oo 

Buel,  R.  H.  Safety  Valves.  (Science  Series  No.  21.) i6mo,  o  50 

Bulmann,  H.  F.,  and  Redmayne,  R.  S.  A.  Colliery  Working  and 

Management 8vo,  6  oo 

Burgh,  N.  P.  Modern  Marine  Engineering 4to,  half  mor.,  10  oo 


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Burstall,  F.  W.    Energy  Diagram  for  Gas.    With  text..8vo,  *i  50 

Diagram  sold  separately *i  oo 

Burt,  W.  A.     Key  to  the  Solar  Compass i6mo,  leather,  2  50 

Burton,   F.   G.      Engineering    Estimates    and   Cost   Accounts. 

i2mo,  *i  50 

Buskett,  E.  W.     Fire  Assaying i2mo,  *i  25 

Byers,    H.  G.,    and   Knight,    H.    G.     Notes   on    Qualitative 

Analysis 8vo,  *i  50 

Cain,  W.     Brief  Course  in  the  Calculus i2mo,  *i  75 

Elastic  Arches.     (Science  Series  No.  48.) i6mo,  o  50 

Maximum  Stresses.     (Science  Series  No.  38.) i6mo,  o  50 

Practical  Designing  Retaining  of  Walls.     (Science  Series 

No.  3.) i6mo,  o  50 

Theory  of  Steel-concrete  Arches  and  of  Vaulted  Structures. 

(Science  Series) i6mo,  o  50 

Theory  of  Voussoir  Arches.    (Science  Series  No.  12.). i6mo,  o  50 

Symbolic  Algebra.     (Science  Series  No.  73.). .......  i6mo,  o  50 

Campin,  F.     The  Construction  of  Iron  Roofs 8vo,  2  oo 

Carpenter,    F.    D.     Geographical    Surveying.     (Science    Series 

No.  37.) i6mo, 

Carpenter,   R.   C.,   and    Diederichs,   H.      Internal -Combustion 

Engines 8vo,  *5  oo 

Carter,  E.  T.     Motive  Power  and  Gearing  for  Electrical  Machin- 
ery  8vo,  *s  oo 

Carter,  H.  A.     Ramie  (Rhea),  China  Grass i2mo,  *2  oo 

Carter,  H.  R.     Modern  Flax,  Hemp,  and  Jute  Spinning 8vo,  *3  oo 

Cathcart,  W.  L.     Machine  Design.     Part  I.  Fastenings 8vo,  *3  oo 

Cathcart,  W.  L.,  and  Chaffee,  J.  I.     Elements  of  Graphic  Statics 

and  General  Graphic  Methods 8vo,  *3  oo 

Short  Course  in  Graphic  Statics i2mo,  *i  50 

Caven,  R.  M.,  and  Lander,  G.  D.     Systematic  Inorganic  Chemis- 
try   i2mo,  *2  oo 

Chalkley,  A.  P.     Diesel  Engines 8vo,  *3  oo 

Chambers'  Mathematical  Tables 8vo,  i  75 

Charnock,  G.  F.     Workshop  Practice.     (Westminster  Series.) 

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Charpentier,  P.     Timber 8vo,  *6  oo 

Chatley,  H.     Principles  and  Designs  of  Aeroplanes.     (Science 

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Chatley,  H.     How  to  Use  Water  Power i2mo,  *i  oo 

Child,  C.  D.     Electric  Arc 8vo  (In  Press.) 

Child,  C.  T.     The  How  and  Why  of  Electricity i2mo,  i  oo 

Christie,  W.  W.      Boiler- waters,  Scale,  Corrosion,    Foaming 

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Water,  Its  Purification  for  Use  in  the  Industries.  .8vo  (In  Press.) 

Church's  Laboratory  Guide.     Rewritten  by  Edward  Kinch.  .8 vo,  *2  50 

Clapperton,  G.     Practical  Papermaking 8vo,  2  50 

Clark,  A.  G.    Motor  Car  Engineering. 

Vol.    I.  Construction 8vo,  *3  oo 

Vol.  II.     Design (In  Press.) 

Clark,  C.  H.    Marine  Gas  Engines i  zmo,  *i  50 

Clark,  D.  K.     Rules,  Tables  and  Data  for  Mechanical  Engineers 

8vo,  5  oo 

Fuel:  Its  Combustion  and  Economy i2mo,  i  50 

The  Mechanical  Engineer's  Pocketbook i6mo,  2  oo 

Tramways:  Their  Construction  and  Working 8vo,  5  oo 

Clark,  J.  M.     New  System  of  Laying  Out  Railway  Turnouts.. 

1 2  mo,  i  oo 
Clausen-Thue,  W.     ABC  Telegraphic  Code.     Fourth  Edition 

1 2 mo,  *5  oo 

Fifth  Edition 8vo,  *7  oo 

The  Ai  Telegraphic  Code 8vo,  *7  50 

Cleemann,  T.  M.     The  Railroad  Engineer's  Practice i2mo,  *i  50 

Clerk,  D.,  and  Well,  F.  E.     Theory  of  the  Gas  Engine.     (Science 

Series  No.  62.) i6mo,  o  50 

Clevenger,  S.  R.     Treatise  on  the  Method  of  Government  Sur- 
veying   i6mo,  mor.,  2  50 

Clouth,  F.     Rubber,  Gutta-Percha,  and  Balata 8vo,  *s  oo 

Cochrane,  J.    Treatise  on  Cement  Specifications 8vo  (In  Press.) 

Coffin,  J.  H.  C.     Navigation  and  Nautical  Astronomy i2mo,  *3  50 

Colburn,  Z.,  and  Thurston,  R.  H.     Steam  Boiler  Explosions. 

(Science  Series  No.  2.) i6mo,  o  50 

Cole,  R.  S.     Treatise  on  Photographic  Optics i2mo,  i  50 

Coles-Finch,  W.     Water,  Its  Origin  and  Use 8vo,  *5  oo 

Collins,  J.  E.     Useful  Alloys  and  Memoranda  for  Goldsmiths, 

Jewelers i6mo,  o  50 


10  D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Constantine,  E.     Marine  Engineers,  Their    Qualifications   and 

Duties 8vo,  *2  oo 

Coombs,  H.  A.     Gear  Teeth.     (Science  Series  No.  120). . .  i6mo,  o  50 

Cooper,  W.  R.     Primary  Batteries 8vo,  *4  oo 

"  The  Electrician  "  Primers 8vo,  *5  oo 

Part  I *i  50 

Part  II *2  50 

Part  III *2  oo 

Copperthwaite,  W.  C.     Tunnel  Shields 4to,  *p  oo 

Corey,  H.  T.     Water  Supply  Engineering 8vo  (In  Press.) 

Corfield,  W.  H.  Dwelling  Houses.  (Science  Series  No.  50.)  i6mo,  o  50 

Water  and  Water-Supply.     (Science  Series  No.  17.). .  i6mo,  o  50 

Cornwall,  H.  B.     Manual  of  Blow-pipe  Analysis 8vo,  *2  50 

Courtney,  C.  F.     Masonry  Dams 8vo,  3  50 

Cowell,  W.  B.     Pure  Air,  Ozone,  and  Water i2mo,  *2  oo 

Craig,  T.     Motion  of  a  Solid  in  a  Fuel.     (Science  Series  No.  49.) 

i6mo,  o  50 

Wave  and  Vortex  Motion.     (Science  Series  No.  43.).  i6mo,  o  50 

Cramp,  W.     Continuous  Current  Machine  Design 8vo,  *2  50 

Crocker,  F.  B.     Electric  Lighting.     Two  Volumes.     8vo. 

Vol.   I.     The  Generating  Plant 3  oo 

Vol.  II.     Distributing  Systems  and  Lamps 3  oo 

Crocker,  F.  B.,  and  Arendt,  M.     Electric  Motors 8vo,  *2  50 

Crocker,  F.  B.,  and  Wheeler,  S.  S.     The  Management  of  Electri- 
cal Machinery i2mo,  *i  oo 

Cross,  C.  F.,  Bevan,  E.  J.,  and  Sindall,  R.  W.     Wood  Pulp  and 

Its  Applications.     (Westminster  Series.) 8vo,  *2  oo 

Crosskey,  L.  R.     Elementary  Prospective 8 vo,  i  oo 

Crosskey,  L.  R.,  and  Thaw,  J.     Advanced  Perspective 8vo,  i  50 

Culley,  J.  L.     Theory  of  Arches.     (Science  Series  No.  87.)i6mo,  o  50 

Davenport,  C.     The  Book.     (Westminster  Series.) 8vo,  *2  oo 

Da  vies,  D.  C.     Metalliferous  Minerals  and  Mining 8vo,  5  oo 

Earthy  Minerals  and  Mining 8vo,  5  oo 

Da  vies,  E.  H.     Machinery  for  Metalliferous  Mines 8vo,  8  oo 

Da  vies,  F.  H.      Electric  Power  and  Traction 8vo,  *2  oo 

Dawson,  P.     Electric  Traction  on  Railways 8vo,  *p  oo 

Day,  C.     The  Indicator  and  Its  Diagrams I2mo,  *2  oo 

Deerr,  N.     Sugar  and  the  Sugar  Cane 8vo,  *8  oc 


D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG    11 

Deite,  C.     Manual  of  Soapmaking.     Trans,  by  S.  T.  King.  .410,  *$  oo 
De  la  Coux,  H.    The  Industrial  Uses  of  Water.     Trans,  by  A. 

Morris 8vo,  *4  50 

Del  Mar,  W.  A.     Electric  Power  Conductors 8vo,  *2  oo 

Denny,  G.  A.     Deep-Level  Mines  of  the  Rand 4to,  *io  oo 

Diamond  Drilling  for  Gold *5  oo 

De  Roos,  J.  D.  C.     Linkages.     (Stience  Series  No.  47.). . .  i6mo,  o  50 

Derr,  W.  L.     Block  Signal  Operation Oblong  i2mo,  *i  50 

Maintenance  of  Way  Engineering (In  Preparation.) 

Desaint,  A.     Three  Hundred  Shades  and  How  to  Mix  Them .  .8vo,  *io  oo 

De  Varona,  A.     Sewer  Gases.     (Science  Series  No.  55.)...i6mo,  o  50 
Devey,  R.  G.     Mill  and  Factory  Wiring.     (Installation  Manuals 

Series.) i2mo,  *i  oo 

Dibdin,  W.  J.     Public  Lighting  by  Gas  and  Electricity 8vo,  *8  oo 

Purification  of  Sewage  and  Water 8vo,  6  50 

Dichman,  C.    Basic  Open-Hearth  Steel  Process 8vo,  *3  50 

Dietrich,  K.     Analysis  of  Resins,  Balsams,  and  Gum  Resins  ,8vo,  *3  oo 
Dinger,  Lieut.  H.  C.     Care  and  Operation  of  Naval  Machinery 

1 2 mo.  *2  oo 

Dixon,  D.  B.     Machinist's'and  Steam  Engineer's  Practical  Cal- 
culator   i6mo,  mor.,  i  25 

Doble,  W.  A.    Power  Plant  Construction  on  the  Pacific  Coast.  (In  Press.) 
Dodd,  G.     Dictionary  of  Manufactures,  Mining,  Machinery,  and 

the  Industrial  Arts i2mo,  i  50 

Dorr,  B.  F.    The  Surveyor's  Guide  and  Pocket  Table-book. 

i6mo,  mor.,  2  oo 

Down,  P  B.     Handy  Copper  Wire  Table i6mo,  *i  oo 

Draper,   C.   H.    Elementary  Text-book   of   Light,   Heat  and 

Sound 1 2010,  i  oo 

Heat  and  the  Principles  of  Thermo-dynamics i2mo,  i  50 

Duckwall,  E.  W.    Canning  and  Preserving  of  Food  Products. 8 vo,  *5  oo 
Dumesny,  P.,  and  Noyer,  J.     Wood  Products,  Distillates,  and 

Extracts 8vo,  *4  50 

Duncan,  W.  G.,  and  Penman,  D.     The  Electrical  Equipment  of 

Collieries 8vo,  *4  oo 

Dunstan,  A.  E.,  and  Thole,  F.  T.  B.    Textbook  of  Practical 

Chemistry I2mo,  *i  40 

Duthie,    A.    L.     Decorative    Glass    Processes.'   (Westminster 

Series) 8vo,  *2  oo 


12    D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATAL3G 

Dwight,  H.  B.    Transmission  Line  Formulas 8vo,  *  (In  Press.) 

Dyson,  S.  S.     Practical  Testing  of  Raw  Materials 8vo,  *s  oo 

Dyson,  S.  S.  and  Clarkson,  S.  S.    Chemical  Works 8vo,  *7  50 

Eccles,  R.G.,  andDuckwall,E.W.  Food  Preservatives.  8 vo,  paper,  o  50 

Eddy,  H.  T.     Researches  in  Graphical  Statics 8vo,  i  50 

Maximum  Stresses  under  Concentrated  Loads 8vo,  i  50 

Edgcumbe,  K.     Industrial  Electrical  Measuring  Instruments . 

8vo,  *2  50 

Eissler,  M.     The  Metallurgy  of  Gold 8vo,  7  50 

—  The  Hydrometallurgy  of  Copper 8vo,  *4  50 

The  Metallurgy  of  Silver 8vo,  4  oo 

The  Metallurgy  of  Argentiferous  Lead 8vo,  5  oc 

Cyanide  Process  for  the  Extraction  of  Gold 8vo,  3  oo 

A  Handbook  of  Modern  Explosives 8vo,  5  oo 

Ekin,  T.  C.      Water  Pipe  and    Sewage    Discharge  Diagrams 

folio,  *3  oo 

Eliot,  C.  W.,  and  Storer,  F.  H.    Compendious  Manual  of  Qualita- 
tive Chemical  Analysis i2mo,  *i  25 

Elliot,  Major  G.  H.     European  Light-house  Systems 8vo,  5  oo 

Ennis,  Wm.  D.     Linseed  Oil  and  Other  Seed  Oils   8vo,  *4  oo 

Applied  Thermodynamics 8vo,  *4  50 

Flying  Machines  To-day i2mo,  *i  50 

Vapors  for  Heat  Engines i2mo,  *i  oo 

Erfurt,  J.     Dyeing  of  Paper  Pulp.     Trans,  by  J.  Hubner. .  .8vo,  *7  50 

Erskine -Murray,  J.     A  Handbook  of  Wireless  Telegraphy.  .8vo,  *3  50 

Evans,  C.  A.     Macadamized  Roads (In  Press.) 

Ewing,  A.  J.     Magnetic  Induction  in  Iron 8vo,  *4  oo 

Fairie,  J.     Notes  on  Lead  Ores i2mo,  *i  oo 

Notes  on  Pottery  Clays i2mo,  *i  50 

Fairley,  W.,  and  Andre,  Geo.  J.     Ventilation  of  Coal  Mines. 

(Science  Series  No.  58.) i6mo,  o  50 

Fairweather,  W.  C.     Foreign  and  Colonial  Patent  Laws  . . .  8vo,  *3  oo 
Fanning,   T.    T.     Hydraulic   and   Water-supply    Engineering. 

8vo,  *5  oo 
Fauth,  P.     The  Moon  in  Modern  Astronomy.     Trans,  by  J. 

McCabe 8vo,  *2  oo 

Fay,  I.  W.    The  Coal-tar  Colors 8vo,  *4  oc 


D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG     13 

Fernbach,  R.  L,    Glue  and  Gelatine. 8vo,  *3  oo 

Chemical  Aspects  of  Silk  Manufacture i2mo,  *i  oo 

Fischer,  E.     The  Preparation  of  Organic  Compounds.     Trans. 

by  R.  V.  Stanford i2mo,  *i  25 

Fish,  J.  C.  L.     Lettering  of  Working  Drawings Oblong  80,  i  oo 

Fisher,  H.  K.  C.,  and  Darby,  W,  Co     Submarine  Cable  Testing. 

8vo,  *3  50 

Fiske,  Lieut.  B.  A.     Electricity  in  Theory  and  Practice 8vo,  2  50 

Fleischmann,  W.     The  Book  of  the  Dairy.     Trans,  by  C.  M. 

Aikman 8vo,  4  oo 

Fleming,    J.    A.     The    Alternate-current    Transformer.     Two 

Volumes 8vo, 

Vol.    I.     The  Induction  of  Electric  Currents *5  oo 

Vol.  II.     The  Utilization  of  Induced  Currents *5  oo 

Propagation  of  Electric  Currents 8vo,  *3  oo 

Fleming,  J,  A.     Centenary  of  the  Electrical  Current 8vo,  *o  50 

—  Electric  Lamps  and  Electric  Lighting 8vo,  *3  oo 

Electric  Laboratory  Notes  and  Forms 4to,  *5  oo 

A  Handbook  for  the  Electrical  Laboratory  and  Testing 

Room.  Two  Volumes 8vo,  each,  *s  oo 

Fluery,  H.  The  Calculus  Without  Limits  or  Infinitesimals. 

Trans,  by  C.  0.  Mailloux (In  Press.) 

Flynn,  P.  J,  Flow  of  Water.  (Science  Series  No«  84.). .  .i6mo,  050 

—  Hydraulic  Tables.     (Science  Series  No.  66.) i6mo,  o  50 

Foley,  N.     British  and  American  Customary  and  Metric  Meas- 
ures   folio,  *3  oo 

Foster,    H.    A.     Electrical    Engineers'    Pocket-book,     (Sixth 

Edition.) I2mo,  leather,  5  oo 

Engineering  Valuation  of  Public  Utilities 8vo,  3  oo* 

Foster,   Gen.   J.   G.     Submarine   Blasting   in   Boston   (Mass.) 

Harbor 410,  3  50 

Fowle,  F.  F.     Overhead  Transmission  Line  Crossings  .. .  .i2mo,  *i  50 

The  Solution  of  Alternating  Current  Problems, 8vo  (In  Press.) 

Fox,  W.  G.     Transition  Curves.     (Science  Series  No.  no.). i6mo,  050 
Fox,  W.,  and  Thomas,  C.  W.     Practical  Course  in  Mechanical 

Drawing i2mo,  i  25 

Foye,  J.  C.     Chemical  Problems,,     (Science  Series  No.  6o..).i6mo,  o  50 

Handbook   of    Mineralogy.      (Science    Series    No.   86.). 

i6mo,  o  50 


14     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Francis,  J.  B.     Lowell  Hydraulic  Experiments 410,  15  oo 

Freudemacher,  P.  W.    Electrical  Mining  Installations.     (In- 
stallation Manuals  Series.) i2mo,  *i  oo 

Frith,  J.    Alternating  Current  Design 8vo,  *2  oo 

Fritsch,  J.     Manufacture  of  Chemical  Manures.    Trans,  by 

D.  Grant 8vo,  *4  oo 

Frye,  A.  I.     Civil  Engineers'  Pocket-book ....  i2mo,  leather,  (In  Press.) 
Fuller,  G.  W.     Investigations  into  the  Purification  of  the  Ohio 

River . 4to,  *io  oo 

Furnell,  J.     Paints,  Colors,  Oils,  and  Varnishes 8vo,  *i  oo 

Gairdner,  J.  W.  I.    Earthwork 8vo  (In  Press.) 

Gant,  L.  W.     Elements  of  Electric  Traction 8vo,  *2  50 

Garforth,  W.  E.     Rules  for  Recovering  Coal  Mines  after  Explo- 
sions and  Fires i2mo,  leather,  i  50 

Gaudard,  J.     Foundations.     (Science  Series  No.  34.) i6mo,  o  50 

Gear,  H.  B.,  and  Williams,  P.  F.     Electric  Central  Station  Dis- 
tributing Systems i2mo,  *3  oo 

Geerligs,  H.  C.  P.     Cane  Sugar  and  Its  Manufacture 8vo,  *s  oo 

Geikie,  J.     Structural  and  Field  Geology 8vo,  *4  oo 

Gerber,  N.     Analysis  of  Milk,  Condensed  Milk,  and  Infants' 

Milk-Food 8vo,  i  25 

Gerhard,  W.  P.     Sanitation,  Water-supply  and  Sewage  Disposal 

of  Country  Houses i2mo,  *2  oo 

Gas  Lighting.     (Science  Series  No.  in.) i6mo,  o  50 

Household  Wastes.     (Science  Series  No.  97.) i6mo,  o  50 

House  Drainage.     (Science  Series  No.  63.) i6mo,  o  50 

Sanitary  Drainage  of  Buildings.     (Science  Series  No.  93.) 

i6mo,  o  50 

Gerhardi,  C.  W.  H.     Electricity  Meters 8vo,  *4  oo 

Geschwind,  L.     Manufacture  of  Alum  and  Sulphates.     Trans. 

by  C.  Salter 8vo,  *s  oo 

Gibbs,  W.  E.     Lighting  by  Acetylene i2mo,  *i  50 

Physics  of  Solids  and  Fluids.     (Carnegie  Technical  Schools 

Text-books.) *i  50 

Gibson,  A.  H.     Hydraulics  and  Its  Application 8vo,  *5  oo 

— —  Water  Hammer  in  Hydraulic  Pipe  Lines i2mo,  *2  oo 

Gilbreth,  F.  B.     Motion  Study.     A  Method  for  Increasing  the 

Efficiency  of  the  Workman I2mo,  *2  oo 


D.  VAN  NO  STRAND  COMPANY'S  SHORT-TITLE  CATALOG    15 

Gilbreth,  F.  B.    Primer  of  Scientific  Management i2mo,  *i  oo 

Gillmore,  Gen.  Q.  A.     Limes,  Hydraulic  Cements  and  Mortars. 

8vo,  4  oo 

—  Roads,  Streets,  and  Pavements i2mo,  2  oo 

Golding,  H.  A.     The  Theta-Phi  Diagram i2mo,  *i  25 

Goldschmidt,  R.     Alternating  Current  Commutator  Motor  .8vo,  *3  oo 

Goodchild,  W.     Precious  Stones.     (Westminster  Series.)- .  -8vo,  *2  oo 

Goodeve,  T.  M.     Textbook  on  the  Steam-engine i2mo,  2  oo 

Gore,  G.     Electrolytic  Separation  of  Metals 8vo,  *3  50 

Gould,  E.  S.     Arithmetic  of  the  Steam-engine i2mo,  i  oo 

—  Calculus.     (Science  Series  No.  112.) i6mo,  o  50 

—  High  Masonry  Dams.     (Science  Series  No.  22.) i6mo,  o  50 

—  Practical  Hydrostatics  and  Hydrostatic  Formulas.     (Science 

Series.) i6mo,       o  50 

Grant,    J.      Brewing    and    Distilling.      (Westminster    Series.) 

8vo   (In  Press.) 

Gratacap,  L.  P.    A  Popular  Guide  to  Minerals.    8vo (In  Press.) 

Gray,  J.     Electrical  Influence  Machines I2mo,       2  oo 

—  Marine  Boiler  Design i2mo  (In  Press.) 

Greenhill,  G.     Dynamics  of  Mechanical  Flight 8vo  (In  Press.) 

Greenwood,  E.     Classified  Guide  to  Technical  and  Commercial 

Books 8vo,  *3  oo 

Gregorius,  R.     Mineral  Waxes.     Trans,  by  C.  Salter i2mo,  *3  oo 

Griffiths,  A.  B.     A  Treatise  on  Manures i2mo,  3  oo 

—  Dental  Metallurgy 8vo,  *3  50 

Gross,  E.     Hops 8vo,  *4  50 

Grossman,  J.     Ammonia  and  its  Compounds i2mo,  *i  25 

Groth,  L.  A.     Welding  and  Cutting  Metals  by  Gases  or  Electric- 
ity  8vo,  *3  oo 

Grover,  F.     Modern  Gas  and  Oil  Engines 8vo,     *2  oo 

Gruner,  A.     Power-loom  Weaving 8vo,     *3  oo 

Giildner,    Hugo.     Internal-Combustion    Engines.      Trans,    by 

H.  Diedrichs 4to,  *io  oo 

Gunther,  C.  0.     Integration i2mo,     *i  25 

Gurden,  R.  L.     Traverse  Tables folio,  half  mor.     *7  50 

Guy,  A.  E.     Experiments  on  the  Flexure  of  Beams 8vo,  *i  25 

Haeder,  H.     Handbook  on  the  Steam-engine.     Trans,  by  H.  H. 

P.  Powles 1 2mo,      3  oo 


16    D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Haenig,  A.    Emery  and  the  Emery  Industry 8vo    (In  Press.) 

Hainbach,  R.     Pottery  Decoration.     Trans,  by  C.  Slater.  .  i2mo,  *3  oo 

Hale,  W.  J.     Calculations  of  General  Chemistry i2mo,  *i  oo 

Hall,  C.  H.     Chemistry  of  Paints  and  Paint  Vehicles i2mo,  *2  oo 

Hall,  R.  H.     Governors  and  Governing  Mechanism i2mo,  *2  oo 

Hall,  W.  S.     Elements  of  the  Differential  and  Integral  Calculus 

8vo,  *2  25 

Descriptive  Geometry 8vo  volume  and  4to  atlas,  *3  50 

Haller,  G.  F.,  and  Cunningham,  E.  T.    The  Tesla  Coil 1 2mo,  *  i  25 

Halsey,  F.  A.     Slide  Valve  Gears i2mo,  i  50 

The  Use  of  the  Slide  Rule.   (Science  Series.) i6mo,  o  50 

Worm  and  Spiral  Gearing.     (Science  Series.) i6mo,  o  50 

Hamilton,  W.  G.     Useful  Information  for  Railway  Men. .  i6mo,  i  oo 
Hammer,  W.  J.     Radium  and  Other  Radioactive  Substances, 

8vo,  *i  oo 

Hancock,  H.     Textbook  of  Mechanics  and  Hydrostatics 8vo,  i  50 

Hardy,  E.     Elementary  Principles  of  Graphic  Statics i2mo,  *i  50 

Harrison,  W.  B.     The  Mechanics'  Tool-book i2mo,  i  50 

Hart,  J.  W.     External  Plumbing  Work 8vo,  *3  oo 

Hints  to  Plumbers  on  Joint  Wiping 8vo,  *3  oo 

Principles  of  Hot  Water  Supply , 8vo,  *3  oo 

Sanitary  Plumbing  and  Drainage 8vo,  *3  oo 

Haskins,  C.  H.     The  Galvanometer  and  Its  Uses i6mo,  i  50 

Hatt,  J.  A.  H.     The  Colorist square  i2mo,  *i  50 

Hausbrand,  E.     Drying  by  Means  of  Air  and  Steam.     Trans. 

by  A.  C.  Wright i2mo,  *2  oo 

Evaporating,  Condensing  and  Cooling  Apparatus.     Trans. 

by  A.  C.  Wright 8vo,  *5  oo 

Hausner,  A.     Manufacture  of  Preserved  Foods  and  Sweetmeats. 

Trans,  by  A.  Morris  and  H.  Robson 8vo,  *3  oo 

Hawke,  W.  H.     Premier  Cipher  Telegraphic  Code 4to,  *5  oo 

—  100,000  Words  Supplement  to  the  Premier  Code 4to,  *5  oo 

Hawkesworth,  T.     Graphical  Handbook  for  Reniforced  Concrete 

Design 4*0,  *2  50 

Hay,  A.     Alternating  Currents 8vo,  *2  50 

—  Electrical  Distributing  Networks  and   Distributing  Lines. 

8vo,  *3  So 

Continuous  Current  Engineering 8vo,  *2  50 

Heap,  Major  D.  P.     Electrical  Appliances 8vo,  2  oo 


D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG     17 

Heaviside,    0.     Electromagnetic    Theory.     Two    volumes. 

8vo,  each,  *5  oo 

Heck,  R.  C.  H.    Steam  Engine  and  Turbine 8vo,  *5  oo 

Steam-Engine  and  Other  Steam  Motors.    Two  Volumes. 

Vol.   I.     Thermodynamics  and  the  Mechanics 8vo,  *3  50 

Vol.  II.     Form,  Construction  and  Working 8vo,  *5  oo 

Notes  on  Elementary  Kinematics 8vo,  boards,  *i  oo 

—  Graphics  of  Machine  Forces 8vo,  boards,  *  i  oo 

Hedges,  K.     Modern  Lightning  Conductors 8vo,  3  oo 

Heermann,  P.     Dyers'    Materials.     Trans,   by   A.  C.  Wright. 

i2mo,  *2  50 
Hellot,  Macquer  and  D'Apligny.     Art  of  Dyeing  Wool,  Silk  and 

Cotton 8vo,  *2  oo 

Henrici,  0.     Skeleton  Structures 8vo,  i  50 

Bering,  D.  W.    Essentials  of  Physics  for  College  Students. 

8vo,  i  75 
Hermann,  G.     The  Graphical  Statics  of  Mechanism.     Trans. 

by  A.  P.  Smith i2mo,  2  oo 

Herring-Shaw,  A.    Domestic  Sanitation  and  Plumbing.  Two 

Parts 8vo,  *5  oo 

Elementary  Science  of  Sanitation  and  Plumbing ....  8vo,  *2  oo 

Herzfeld,  J.     Testing  of  Yarns  and  Textile  Fabrics 8.vo,  *3  50 

Hildebrandt,  A.     Airships,  Past  and  Present. 8vo,  *3  50 

Hildenbrand,  B.  W.     Cable-Making.      (Science  Series  No.  32.) 

i6mo,  o  50 

Hildich,  H.     Concise  History  of  Chemistry i2mo,  *i  25 

Hill,  J.  W.     The  Purification  of  Public  Water  Supplies.     New 

Edition , (In  Press.) 

—  Interpretation  of  Water  Analysis (In  Press.) 

Hiroi,  I.     Plate  Girder  Construction.     (Science  Series  No.  95.) 

i6mo,  o  50 

Statically-Indeterminate  Stresses i2mo,  *2  oo 

Hirshfeld,    C.    F.      Engineering     Thermodynamics.     (Science 

Series.) i6mo,  o  50 

Hobart,  H.  M.     Heavy  Electrical  Engineering 8vo,  *4  50 

Design  of  Static  Transformers 8vo,  *2  oo 

• Electricity 8vo,  *2  oo 

Electric  Trains 8vo,  *2  50 

—  Electric  Propulsion  of  Ships 8vo,  *2  oo 


18    D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Hobart,  J.  F.    Hard  Soldering,  Soft  Soldering,  and  Brazing . 

i2mo,  (In  Press.) 
Hobbs,  W.  R.  P.     The  Arithmetic  of  Electrical  Measurements 

i2mo,  o  50 

Hoff,  J.  N.     Paint  and  Varnish  Facts  and  Formulas. ....  i2mo,  *i  50 
Hoff,  Com.W.  B.  The  Avoidance  of  Collisions  at  Sea.  i6mo,  mor.,     o  75 

Hole,  W.     The  Distribution  of  Gas 8vo,  *7  50 

Holley,  A.  L.     Railway  Practice folio,  12  oo 

Holmes,  A.  B.     The  Electric  Light  Popularly  Explained. 

i2mo,  paper,  o  50 

Hopkins,  N.  M.     Experimental  Electrochemistry 8vo,  *3  oo 

Model  Engines  and  Small  Boats i2mo,  i  25 

Hopkinson,  J.,  Shoolbred,  J.  N.,  and  Day,  R.  E.     Dynamic 

Electricity.     (Science  Series  No.  71.) i6mo,  o  50 

Horner,  J.     Engineers'  Turning 8vo,  *3  50 

Metal  Turning i2mo,  i  50 

Toothed  Gearing i2mo,  2  25 

Houghton,  C.  E.    The  Elements  of  Mechanics  of  Materials.  i2mo,  *2  oo 

Houllevigue,  L.     The  Evolution  of  the  Sciences 8vo,  *2  oo 

Howe,  G.     Mathematics  for  the  Practical  Man i2mo,  *i  25 

Howorth,  J.     Repairing  and  Riveting  Glass,  China  and  Earthen- 
ware  8vo,  paper,  *o  50 

Hubbard,  E.     The  Utilization  of  Wood-waste 8vo,  *2  50 

Hubner,  J.    Bleaching  and  Dyeing  of  Vegetable  and  Fibrous 

Materials.     (Outlines  of  Industrial  Chemistry.) .  *(In  Press.) 
Hudson,    O.    F.    Iron    and    Steel.     (Outlines    of    Industrial 

Chemistry.) 8vo    (In  Press.) 

Humber,  W.     Calculation  of  Strains  in  Girders i2mo,  2  50 

Humphreys,    A.    C.     The    Business    Features    of   Engineering 

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Hunter,  A.    Bridge  Work 8vo    (In  Press.) 

Hurst,  G.  H.     Handbook  of  the  Theory  of  Color 8vo,  *2  50 

Dictionary  of  Chemicals  and  Raw  Products 8vo,  *3  oo 

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Hurst,  H.  E.,  and  Lattey,  R.  T.     Text-book  of  Physics 8vo,  *3  oo 

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D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG     19 

Hutchinson,  R.  W.,  Jr.,  and  Ihlseng,  M.  C.  Electricity  in 

Mining i2mo  (In  Press.) 

Hutchinson,  W.  B.  Patents  and  How  to  Make  Money  Out  of 

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Hutton,  W.  S.     Steam-boiler  Construction 8vo,  6  oo 

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Hyde,  E.  W.     Skew  Arches.     (Science  Series  No.  15.) i6mo,  050 

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Ingle,  H.     Manual  of  Agricultural  Chemistry.... 8vo,  *3  oo 

Innes,  C.  H.     Problems  in  Machine  Design I2mo,  *2  oo 

Air  Compressors  and  Blowing  Engines i2mo,  *2  oo 

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Ivatts,  E.  B.    Railway  Management  at  Stations 8vo,  *2  50 

Jacob,  A.,  and  Gould,  E.  S.     On  the  Designing  and  Construction 

of  Storage  Reservoirs.     (Science  Series  No.  6.).  .i6mo,  o  50 

Jamieson,  A.     Text  Book  on  Steam  and  Steam  Engines. .  .  .  8vo,  3  oo 

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Jannettaz,  E.     Guide  to  the  Determination  of  Rocks.     Trans. 

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Jennings,   A.   S.     Commercial   Paints   and   Painting.     (West- 
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Jennison,  F.  H.     The  Manufacture  of  Lake  Pigments 8vo,  *3  oo 

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Jockin,  W.     Arithmetic  of  the  Gold  and  Silversmith i2mo,  *i  oo 

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20     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Johnson,  W.  McA.     The  Metallurgy  of  Nickel (In  Preparation.) 

Johnston,  J.  F.  W.,  and  Cameron,  C.     Elements  of  Agricultural 

Chemistry  and  Geology i2mo,  2  60 

Joly,  J.     Radioactivity  and  Geology i2mo,  *3  oo 

Jones,  H.  C.     Electrical  Nature  of  Matter  and  Radioactivity 

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Jones,  M.  W.     Testing  Raw  Materials  Used  in  Paint i2mo,  *2  oo 

Jones,  L.,  and  Scard,  F.  I.     Manufacture  of  Cane  Sugar 8vo,  *5  oo 

Jordan,  L.  C.    Practical  Railway  Spiral . .  .  i2mo,  Leather  *(In  Press.) 
Joynson,  F.  H.     Designing  and  Construction  of  Machine  Gear- 
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Jiiptner,  H.  F.  V.    Siderology:  The  Science  of  Iron 8vo,  *5  oo 

Kansas  City  Bridge 4to,  6  oo 

Kapp,  G.     Alternate  Current  Machinery.     (Science  Series  No. 

96.) i6mo,  o  50 

Electric  Transmission  of  Energy i2mo,  3  50 

Keim,  A.  W.     Prevention  of  Dampness  in  Buildings 8vo,  *2  oo 

Keller,  S.  S.    Mathematics  for  Engineering  Students. 

i2mo,  half  leather, 

Algebra  and  Trigonometry,  with  a  Chapter  on  Vectors. ...  *i  75 

Special  Algebra  Edition *i  oo 

Plane  and  Solid  Geometry *i  25 

Analytical  Geometry  and  Calculus *2  oo 

Kelsey,  W.   R.      Continuous-current    Dynamos  and  Motors. 

8vo,  *2  50 
Kemble,  W.  T.,  and  Underbill,  C.  R.     The  Periodic  Law  and  the 

Hydrogen  Spectrum 8vo,  paper,  *o  50 

Kemp,  J.  F.     Handbook  of  Rocks 8vo,  *i  50 

Kendall,  E.    Twelve  Figure  Cipher  Code 4to,  *is  oo 

Kennedy,   A.   B.   W.,  and  Thurston,   R.   H.     Kinematics   of 

Machinery.     (Science  Series  No.  54.) i6mo,  o  50 

Kennedy,  A.  B.  W.,  Unwin,  W.  C.,  and  Idell,  F.  E.    Compressed 

Air.     (Science  Series  No.  106.) i6mo,  o  50 

Kennedy,  R.     Modern  Engines  and  Power  Generators.     Six 

Volumes 4to,  15  oo 

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Kennelly,  A.  E.     Electro-dynamic  Machinery 8vo,  I  50 

Kent,  W.     Strength  of  Materials.     (Science  Series  No.  41.).  i6mo,  050 

Kershaw,  J.  B.  C.     Fuel,  Water  and  Gas  Analysis 8vo,  *2  50 

—  Electrometallurgy.     (Westminster  Series.) 8vo,  *2  oo 

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Kinzbrunner,  C.     Alternate  Current  Windings 8vo,  *i  50 

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—  Testing  of  Alternating  Current  Machines 8vo,  *2  oo 

Kirkaldy,    W.    G.     David    Kirkaldy's   System    of    Mechanical 

Testing 4to,  10  oo 

Kirkbride,  J.     Engraving  for  Illustration 8vo,  *i  50 

Kirkwood,  J.  P.     Filtration  of  River  Waters 4to,  7  50 

Klein,  J.  F.     Design  of  a  High  speed  Steam-engine 8vo,  *5  oo 

Physical  Significance  of  Entropy 8vo,  *i  50 

Kleinhans,  F.  B.     Boiler  Construction 8vo,  3  oo 

Knight,  R.-Adm.  A.  M.     Modern  Seamanship 8vo,  *7  50 

Half  Mor.  *9  oo 

Knox,  W.  F.     Logarithm  Tables (In  Preparation.) 

Knott,  C.  G.,  and  Mackay,  J.  S.     Practical  Mathematics .  .  .  8vo,  2  oo 

Koester,  F.     Steam-Electric  Power  Plants 4to,  *5  oo 

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Koller,  T.     The  Utilization  of  Waste  Products 8vo,  *3  50 

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Lambert,  T.     Lead  and  its  Compounds 8vo,  *3  50 

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Larner,  E.  T.     Principles  of  Alternating  Currents i2mo,  *i  25 

Larrabee,   C.   S.     Cipher  and  Secret  Letter  and  Telegraphic 

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22    D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

La  Rue,  B.  F.     Swing  Bridges.     (Science  Series  No.  107.) .  i6mo,  o  50 
Lassar-Cohn,  Dr.     Modern  Scientific  Chemistry.     Trans,  by  M. 

M.  Pattison  Muir I2mo,  *2  oo 

Latimer,  L.  H.,  Field,  C.  J.,  and  Howell,  J.  W.     Incandescent 

Electric  Lighting.     (Science  Series  No.  57.) i6mo,  o  50 

Latta,  M.  N.     Handbook  of  American  Gas-Engineering  Practice. 

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Leask,  A.  R.     Breakdowns  at  Sea i2mo,  2  oo 

Refrigerating  Machinery i2mo,  2  oo 

Lecky,  S.  T.  S.     "  Wrinkles  "  in  Practical  Navigation  .....  8vo,  *8  oo 
Le  Doux,  M.     Ice-Making  Machines.     (Science  Series  No.  46.) 

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Leeds,  C.  C.    Mechanical  Drawing  for  Trade  Schools .  oblong,  4to, 

High  School  Edition *i  25 

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Lefe*vre,  L.     Architectural  Pottery.     Trans,  by  H.  K.  Bird  and 

W.  M.  Binns 4to,  *7  50 

Lehner,  S.     Ink  Manufacture.     Trans,  by  A.  Morris  and  H. 

Robson 8vo,  *2  50 

Lemstrom,  S.     Electricity  in  Agriculture  and  Horticulture .  .  8vo,  *  i  50 
Le  Van,  W.  B.     Steam-Engine  Indicator      (Science  Series  No. 

78.) ; i6mo,  o  50 

Lewes,  V.  B.     Liquid  and  Gaseous  Fuels.     ^Westminster  Series.) 

8vo,  *2  oo 

Lewis,  L.  P.    Railway  Signal  Engineering 8vo,  *3  50 

Lieber,  B.  F.     Lieber's  Standard  Telegraphic  Code 8vo,  *io  oo 

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Uvermore,  V.  P.,  and  Williams,  J.     How  to  Become  a  Com- 
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Livingstone,    R.    Design  and  Construction  of  Commutators. 

8vo,     *2  25 

Lobben,  P.    Machinists'  and  Draftsmen's  Handbook 8vo,      2  50 

Locke,  A.  G.  and  C.  G.     Manufacture  of  Sulphuric  Acid 8vo,     10  oo 

Lockwood,  T.  D.  Electricity,  Magnetism,  and  Electro-teleg- 
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Electrical  Measurement  and  the  Galvanometer i2mo,      o  75 

Lodge,  0.  J.     Elementary  Mechanics I2mo,       i  50 

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Loewenstein,  L.  C.,  and  Crissey,  C.  P.    Centrifugal  Pumps .  8vo,    *4  50 

Lord,  R.  T.     Decorative  and  Fancy  Fabrics 8vo,     *3  50 

Loring,  A.  E.     A  Handbook  of  the  Electromagnetic  Telegraph. 

(Science  Series  No.  39) i6mo,      o  50 

Lubschez,  B.  J.     Perspective (In  Press.) 

Lucke,  C.  E.     Gas  Engine  Design :  .8vo,     *3  oo 

Power  Plants:  their  Design,  Efficiency,  and  Power  Costs. 

2  vols (In  Preparation.) 

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Technical  Chemists'  Handbook I2mo,  leather,     *3  50 

Lunge,  G.  Technical  Methods  of  Chemical  Analysis.  Trans, 
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specialists. 

Vol.   I.     In  two  parts ,. ., 8vo,  "15  oo 

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Lupton,  A.,  Parr,  G.  D.  A.,  and  Perkin,  H.    Electricity  as  Applied 

to  Mining 8vo,     *4  50 

Luquer,  L.  M.     Minerals  in  Rock  Sections 8vo,     *i  50 

Macewen,  H.  A.     Food  Inspection 8vo,     *2  50 

Mackenzie,  N.  F.     Notes  on  Irrigation  Works 8vo,     *2  50 

Mackie,  J.     How  to  Make  a  Woolen  Mill  Pay 8vo,     *2  oo 


24     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

Mackrow,    C.     Naval    Architect's    and    Shipbuilder's    Pocket- 
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Maguire,  Wm.  R.     Domestic  Sanitary  Drainage  and  Plumbing 

8vo,  4  oo 
Mallet,    A.     Compound    Engines.     Trans,    by    R.    R.    Duel. 

(Science  Series  No.  10.) i6mo, 

Mansfield,  A.  N.     Electro-magnets.     (Science  Series  No.  64) 

i6mo,  o  50 
Marks,  E.  C.  R.     Construction  of  Cranes  and  Lifting  Machinery 

I2mo,  *i  50 

Construction  and  Working  of  Pumps I2mo,  *i  50 

Manufacture  of  Iron  and  Steel  Tubes I2mo,  *2  oo 

Mechanical  Engineering  Materials I2mo,  *i  oo 

Marks,  G.  C.     Hydraulic  Power  Engineering 8vo,  3  50 

Inventions,  Patents  and  Designs 12 mo,  *i  oo 

Marlow,  T.  G.     Drying  Machinery  and  Practice 8vo,  *5  oo 

Marsh,  C.  F.     Concise  Treatise  on  Reinforced  Concrete.. .  .8vo,  *2  50 

Reinforced    Concrete    Compression    Member   Diagram.  i  50 

Marsh,  C.  F.,  and  Dunn,  W.     Reinforced  Concrete 4to,  *S  oo 

Manual  of  Reinforced  Concrete  and  Concrete  Block  Con- 
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Marshall,  W.  J.,  and  Sankey,  H.  R.    Gas  Engines.    (Westminster 

Series.) 8vo,  *2  oo 

Martin,   G.    Triumphs  and  Wonders  of  Modern  Chemistry. 

8vo,  *2  oo 

Martin,  N.     Reinforced  Concrete (In  Press.) 

Massie,  W.  W.,  and  Underbill,  C.  R.    Wireless  Telegraphy  and 

Telephony I2mo,  *i  oo 

Matheson,  D.     Australian  Saw-Miller's  Log  and  Timber  Ready 

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Mathot,  R.  E.     Internal  Combustion  Engines 8vo,  *6  oo 

Maurice,  W.     Electric  Blasting  Apparatus  and  Explosives  ..8vo,  *3  50 

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Maxwell,  J.  C.     Matter  and  Motion.     (Science  Series  No.  36.) 

i6mo,  o  50 
Maxwell,  W.  H.,  and  Brown,  J.  T.     Encyclopedia  of  Municipal 

and  Sanitary  Engineering 4to,  *io  oo 

Mayer,  A.  M.     Lecture  Notes  on  Physics 8vo,  2  oo 

•McCullough,  R.  S.     Mechanical  Theory  of  Heat 8vo,  3  50 

Mclntosh,  J.  G.     Technology  of  Sugar 8vo,  *4  So 


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Mclntosh,  J.  G.     Industrial  Alcohol 8vo,  *3  oo 

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Vol.  I.     Oil  Crushing,  Refining  and  Boiling *3  5<> 

Vol.  II.     Varnish  Materials  and  Oil  Varnish  Making *4  oo 

Vol.  in.     Spirit  Varnishes  and  Materials *4  5<> 

McKnight,  J.  D.,  and  Brown,  A.  W.     Marine  Multitubular 

Boilers.  . .-. *i  50 

McMaster,  J.  B.     Bridge  and  Tunnel  Centres.     (Science  Series 

No.  20.) i6mo,  o  50 

McMechen,  F.  L.     Tests  for  Ores,  Minerals  and  Metals.. .  i2mo,  *i  oo 

McNeill,  B.     McNeill's  Code 8vo,  *6  oo 

McPherson,  J.  A.     Water-works  Distribution 8vo,  2  50 

Melick,  C.  W.     Dairy  Laboratory  Guide I2mo,  *i  25 

Merck,  E.     Chemical  Reagents ;  Their  Purity  and  Tests.. .  .8vo,  *i  50 

Merritt,  Wm.  H.  Field  Testing  for  Gold  and  Silver .  i6mo,  leather,  i  50 

Messer,  W.  A.     Railway  Permanent  Way 8vo    (In  Press.) 

Meyer,  J.  G.  A.,  and  Pecker,  C.  G.     Mechanical  Drawing  and 

Machine  Design 4to,  5  oo 

Michell,  S.     Mine  Drainage 8vo,  10  oo 

Mierzinski,  S.     Waterproofing  of  Fabrics.     Trans,  by  A.  Morris 

and  H.  Robson 8vo,  *2  50 

Miller,  E.  H.     Quantitative  Analysis  for  Mining  Engineers ..  8vo,  *i  50 
Miller,  G.  A.     Determinants.     (Science  Series  No.  105.).  .16010, 

Milroy,  M.  E.  W.     Home  Lace-making i2mo,  *i  oo 

Minifie,  W.     Mechanical  Drawing 8vo,  *4  oo 

Mitchell,  C.  A.,  and  Prideaux,  R.  M.     Fibres  Used  in  Textile  and 

Allied  Industries 8vo,  *3  oo 

Modern  Meteorology I2mo,  i  50 

Monckton,  C.  C.  F.     Radiotelegraphy.     (Westminster  Series.) 

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Spanish-English  Technical  Terms 64mo,  leather,  *  i  oo 

Moore,  E.  C.  S.     New  Tables  for  the  Complete  Solution  of 

Ganguillet  and  Kutter's  Formula 8vo,  *5  oo 

Morecroft,  J.  H.,  and  Hehre,  F.  W.    Testing  Electrical  Ma- 
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Moreing,  C.  A.,  and  Neal,  T.     New  General  and  Mining  Tele- 
graph Code 8vo,  *s  oo 


26     D.  VAX  NOSTRAND  COMPANY'S  SHORT  TITLE  CATALOG 

Morgan,  A.  P.     Wireless  Telegraph  Construction  for  Amateurs. 

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Moses,  A.  J.     The  Characters  of  Crystals 8vo,  *2  oo 

Moses,  A.  J.,  and  Parsons,  C.  I.  Elements  of  Mineralogy.  .8vo,  *2  50 
Moss,  S.  A.  Elements  of  Gas  Engine  Design.  (Science 

Series.) ibmo,  o  50 

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Mulford,  A.  C.    Boundaries  and  Landmarks  .....* (In  Press.) 

Mullin,  J.  P.     Modern  Moulding  and  Pattern-making.  .  .  .  i2mo,  2  50 
Munby,  A.  E.     Chemistry  and  Physics  of  Building  Materials. 

(Westminster  Series.) 8vo,  *2  oo 

Murphy,  J.  G.     Practical  Mining i6mo,  i  oo 

Murray,  J.  A.     Soils  and  Manures.     (Westminster  Series.). 8 vo,  *2  oo 

Naquet,  A.  Legal  Chemistry I2mo,  2  oo 

Nasmith,  J.  The  Student's  Cotton  Spinning 8vo,  3  oo 

—  Recent  Cotton  Mill  Construction i2mo,  2  oo 

Neave,  G.  B.,  and  Heilbron,  I.  M.  Identification  of  Organic 

Compounds i2mo,  *i  25 

Neilson,  R.  M.  Aeroplane  Patents 8vo,  *2  oo 

Nerz,  F.  Searchlights.  Trans,  by  C.  Rodgers 8vo,  *3  oo 

Nesbit,  A.  F.  Electricity  and  Magnetism (In  Preparation.) 

Neuberger,  H.,  and  Noalhat,  H.  Technology  of  Petroleum. 

Trans,  by  J.  G.  Mclntosh 8vo,  *io  oo 

Newall,  J.  W.  Drawing,  Sizing  and  Cutting  Bevel-gears.  .8vo,  i  50 

Nicol,  G.  Ship  Construction  and  Calculations 8vo,  *4  50 

Nipher,  F.  E.  Theory  of  Magnetic  Measurements i2mo,  i  oo 

Nisbet,  H.  Grammar  of  Textile  Design 8vo,  *3  oo 

Nolan,  H.  The  Telescope.  (Science  Series  No.  51.) i6mo,  o  50 

Noll,  A.  How' to  Wire  Buildings i2mo,  i  50 

Nugent,  E.  Treatise  on  Optics i2mo,  i  50 

O'Connor,  H.     The  Gas  Engineer's  Pocketbook.  ..  i2mo,  leather,  350 

—  Petrol  Air  Gas i2mo,  *o  75 

Ohm,  G.  S.,  and  Lockwood,  T.  D.     Galvanic  Circuit.     Trans,  by 

William  Francis.  (Science  Series  No.  102.).  .  . .  i6mo,  o  50 

Olsen,  J.  C.  Text  book  of  Quantitative  Chemical  Analysis .  .8vo,  *4  oo 
Olsson,  A.  Motor  Control,  in  Turret  Turning  and  Gun  Elevating. 

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Pamely,  C.     Colliery  Manager's  Handbook 8vo,  *io  oo 

Parr,  G.  D.  A.     Electrical  Engineering  Measuring  Instruments. 

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Parry,  L.  A.     Risk  and  Dangers  of  Various  Occupations 8vo,  *3  oo 

Parshall,  H.  F.,  and  Hobart,  H.  M.     Armature  Windings  ....  4to,  *7  50 

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Parshall,  H.  F.,  and  Parry,  E.     Electrical  Equipment  of  Tram- 
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Partington,  J.  R.    Higher  Mathematics  for  Chemical  Students 

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Rankine,  W.  J.  M.,  and  Bamber,  E.  F.     A  Mechanical  Text- 
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D.  VAN  NOSTRAND  COMPANY^  SHORT-TITLE  CATALOG    31 

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Robinson,  J.  B.     Architectural  Composition 8vo,  *2  50 

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Rogers,  A.,  and  Aubert,  A.  B.     Industrial  Chemistry 8vo,  *5  oo 

Rogers,  F.     Magnetism  of  Iron  Vessels.     (Science  Series  No.  30.) 

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Rowan,  F.  J.   Practical  Physics  of  the  Modern  Steam-boiler.Svo,  7  50 
Rowan,  F.  J.,  and  Idell,  F.  E.     Boiler  Incrustation  and  Corro- 
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Roxburgh,  W.     General  Foundry  Practice 8vo,  *3  50 

Ruhmer,    E.     Wireless    Telephony.     Trans,    by    J.    Erskine- 

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Sabine,  R.  History  and  Progress  of  the  Electric  Telegraph.  i2mo,  i  25 

Saeltzer,  A.     Treatise  on  Acoustics I2mo,  i  oo 

Salomons,  D.     Electric  Light  Installations.     i2mo. 

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Saunnier,  C.     Watchmaker's  Handbook i2mo,  3  oo 

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Scheele,  C.  W.     Chemical  Essays 8vo,  *2  oo 

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Schmall,  C.  N.     First  Course  in  Analytic  Geometry,  Plane  and 

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Schumann,  F.     A  Manual  of  Heating  and  Ventilation. 

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D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG    33 

Scott,  W.  W.     Qualitative  Chemical  Analysis.     A  Laboratory 

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Searle,  G.  M.     "  Sumners'  Method."     Condensed  and  Improved. 

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Sellew,  W.  H.     Steel  Rails 4to  (7n  Press.) 

Senter,  G.     Outlines  of  Physical  Chemistry I2mo,  *i  75 

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Sexton,  A.  H.     Fuel  and  Refractory  Materials I2mo,  *2  50 

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Seymour,  A.     Practical  Lithography 8vo,  *2  50 

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Shaw,  H.  S.  H.     Mechanical  Integrators.    (Science  Series  No. 

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Shaw,  P.  E.     Course  of  Practical  Magnetism  and  Electricity .  8vo,  *i  oo 

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34     D.  VAN  NOSTRAND  COMPANY'S  SHORT-TITLE  CATALOG 

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Shunk,  W.  F.     The  Field  Engineer i2mo,  mor.,  2  50 

Simmons,  W.  H.,  and  Appleton,  H.  A.     Handbook  of  Soap 

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Simms,  F.  W.     The  Principles  and  Practice  of  Leveling 8vo,  2  50 

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Simpson,  G.     The  Naval  Constructor i2mo,  mor.,  *5  oo 

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Sinclair,  A.     Development  of  the  Locomotive  Engine. 

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Sindall,  R.  W.     Manufacture  of  Paper.     (Westminster  Series.) 

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Sloane,  T.  O'C.     Elementary  Electrical  Calculations i2mo,  *2  oo 

Smith,  C.A.  M.    Handbook  of  Testing.    Vol.1.     Materials..  *2  50 
Smith,  C.  A.  M.,  and  Warren,  A.  G.    New  Steam  Tables .  8vo, 

Smith,  C.  F.     Practical  Alternating  Currents  and  Testing ..  8vo,  *2  50 

Practical  Testing  of  Dynamos  and  Motors 8vo,  *2  oo 

Smith,  F.  E.     Handbook  of  General  Instruction  for  Mechanics. 

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Smith,  J.  C.     Manufacture  of  Paint. 8vo,  *3  oo 

Smith,  R.  H.    Principles  of  Machine  Work I2mo,  *3  oo 

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Smith,  W.     Chemistry  of  Hat  Manufacturing i2mo,  *3  oo 

Snell,  A.  T.     Electric  Motive  Power 8vo,  *4  oo 

Snow,  W.  G.     Pocketbook  of  Steam  Heating  and  Ventilation. 

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Snow,  W.  G.,  and  Nolan,  T.     Ventilation  of  Buildings.     (Science 

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Soddy,  F.     Radioactivity 8vo,  *3  oo 

Solomon,  M.     Electric  Lamps.     (Westminster  Series.) 8vo,  *2  oo 

Sothern,  J.  W.     The  Marine  Steam  Turbine 8vo,  *5  oo 

Southcombe,  J.  E.     Paints,  Oils,  and  Varnishes.     (Outlines  of 

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D.  VAN  NOSTKAND  COMPANY'S  SHORT-TITLE  CATALOG  35 

Soxhlet,  D.  H.     Dyeing  and  Staining  Marble.     Trans,  by  A. 

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Spang,  H.  W.     A  Practical  Treatise  on  Lightning  Protection. 

1 2 mo,  i  oo 
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Shreve.     (Science  Series  No.  23.) i6mo,  o  50 

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